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In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that:

x^{2} \equiv q (\mod n) .

Otherwise, q is called a quadratic nonresidue modulo n.

Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.

History, conventions, and elementary facts

Fermat, Euler, Lagrange, Legendre, and other number theorists of the 17th and 18th centuries proved some theorems^[1] and made some conjectures^[2] about quadratic residues, but the first systematic treatment is § IV of Gauss's Disquisitiones Arithmeticae (1801). Article 95 introduces the terminology "quadratic residue" and "quadratic nonresidue", and states that, if the context makes it clear, the adjective "quadratic" may be dropped.

For a given n a list of the quadratic residues modulo n may be obtained by simply squaring the numbers 0, 1, …, n − 1. Because a² ≡ (n − a)² (mod n), the list of squares modulo n is symmetrical around n/2, and the list only needs to go that high. This can be seen in the table below.

Thus, the number of quadratic residues modulo n cannot exceed n/2 + 1 (n even) or (n + 1)/2 (n odd).^[3]

The product of two residues is always a residue.

Prime modulus

Modulo 2, every integer is a quadratic residue.

Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues. In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field Z/pZ. (In other words, every congruence class except zero modulo p has a multiplicative inverse. This is not true for composite moduli.)^[4]

Following this convention, the multiplicative inverse of a residue is a residue, and the inverse of a nonresidue is a nonresidue.^[5]

Following this convention, modulo a prime number there are an equal number of residues and nonresidues.^[4]

Modulo a prime, the product of two nonresidues is a residue and the product of a nonresidue and a (nonzero) residue is a nonresidue.^[5]

The first supplement^[6] to the law of quadratic reciprocity is that if p ≡ 1 (mod 4) then −1 is a quadratic residue modulo p, and if p ≡ 3 (mod 4) then −1 is a nonresidue modulo p. This implies the following:

If p ≡ 1 (mod 4) the negative of a residue modulo p is a residue and the negative of a nonresidue is a nonresidue.

If p ≡ 3 (mod 4) the negative of a residue modulo p is a nonresidue and the negative of a nonresidue is a residue.

Prime power modulus

All odd squares are ≡ 1 (mod 8) and a fortiori ≡ 1 (mod 4). If a is an odd number and m = 8, 16, or some higher power of 2, then a is a residue modulo m if and only if a ≡ 1 (mod 8).^[7]

For example, mod (32) the odd squares are
1² ≡ 15² ≡ 1

3² ≡ 13² ≡ 9

5² ≡ 11² ≡ 25

7² ≡ 9² ≡ 17

and the even ones are

0² ≡ 8² ≡ 16² ≡ 0

2² ≡ 6²≡ 10² ≡ 14²≡ 4

4² ≡ 12² ≡ 16.

So a nonzero number is a residue mod 8, 16, etc., if and only if it is of the form 4^k(8n + 1).

A number A relatively prime to an odd prime p is a residue modulo any power of p if and only if it is a residue modulo p.^[8]

If the modulus is pⁿ,

then p^kA

is a residue modulo pⁿ if k ≥ n

is a nonresidue modulo pⁿ if k < n is odd

is a residue modulo pⁿ if k < n is even and A is a residue

is a nonresidue modulo pⁿ if k < n is even and A is a nonresidue.^[9]

Notice that the rules are different for powers of two and powers of odd primes.

Modulo an odd prime power n = p^k, the products of residues and nonresidues relatively prime to p obey the same rules as they do mod p; p is a nonresidue, and in general all the residues and nonresidues obey the same rules, except that the products will be zero if the power of p in the product ≥ n.

Modulo 8, the product of the nonresidues 3 and 5 is the nonresidue 7, and likewise for permutations of 3, 5 and 7. In fact, the multiplicative group of the non-residues and 1 form the Klein four-group.

Composite modulus not a prime power

The basic fact in this case is

if a is a residue modulo n, then a is a residue modulo p^k for every prime power dividing n.

if a is a nonresidue modulo n, then a is a nonresidue modulo p^k for at least one prime power dividing n.

Modulo a composite number, the product of two residues is a residue. The product of a residue and a nonresidue may be a residue, a nonresidue, or zero.

For example, from the table for modulus 6 1, 2, 3, 4, 5 (residues in bold).
The product of the residue 3 and the nonresidue 5 is the residue 3, whereas the product of the residue 4 and the nonresidue 2 is the nonresidue 2.

Also, the product of two nonresidues may be either a residue, a nonresidue, or zero.

For example, from the table for modulus 15 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (residues in bold).
The product of the nonresidues 2 and 8 is the residue 1, whereas the product of the nonresidues 2 and 7 is the nonresidue 14.

This phenomenon can best be described using the vocabulary of abstract algebra. The congruence classes relatively prime to the modulus are a group under multiplication, called the group of units of the ring Z/nZ, and the squares are a subgroup of it. Different nonresidues may belong to different cosets, and there is no simple rule that predicts which one their product will be in. Modulo a prime, there is only the subgroup of squares and a single coset.

The fact that, e.g., modulo 15 the product of the nonresidues 3 and 5, or of the nonresidue 5 and the residue 9, or the two residues 9 and 10 are all zero comes from working in the full ring Z/nZ, which has zero divisors for composite n.

For this reason some authors^[10] add to the definition that a quadratic residue q must not only be a square but must also be relatively prime to the modulus n.

Although it makes things tidier, this article does not insist that residues must be coprime to the modulus.

Notations

Gauss^[11] used R and N to denote residuacity and non-residuacity, respectively;

for example, 2 R 7 and 5 N 7, or 1 R 8 and 3,5,7 N 8.

Although this notation is compact and convenient for some purposes,^[12]^[13] a more useful notation is the Legendre symbol, also called the quadratic character, which is defined for all integers a and positive odd prime numbers p as

(\frac{a}{p}) = {\begin{cases} 0 if p divides a \\ + 1 if a R p and p does not divide a \\ - 1 if a N p . \end{cases}

There are two reasons why numbers ≡ 0 (mod p) are treated specially. As we have seen, it makes many formulas and theorems easier to state. The other (related) reason is that the quadratic character is a homomorphism from the multiplicative group of nonzero conguence classes modulo p to the complex numbers. Setting $(\frac{n p}{p}) = 0$ allows its domain to be extended to the multiplicative semigroup of all the integers.^[14]

One advantage of this notation over Gauss's is that the Legendre symbol is a function that can be used in formulas.^[15] It can also easily be generalized to cubic, quartic and higher power residues.^[16]

There is a generalization of the Legendre symbol for composite values of p, the Jacobi symbol, but its properties are not as simple: if m is composite and the Jacobi symbol $(\frac{a}{m}) = - 1,$ then a N m, and if a R m then $(\frac{a}{m}) = 1,$ but if $(\frac{a}{m}) = 1$ we do not know whether a R m or a N m. If m is prime, the Jacobi and Legendre symbols agree.

Distribution of quadratic residues

Although quadratic residues appear to occur in a rather random pattern modulo n, and this has been exploited in such applications as acoustics and cryptography, their distribution also exhibits some striking regularities.

Using Dirichlet's theorem on primes in arithmetic progressions, the law of quadratic reciprocity, and the Chinese remainder theorem (CRT) it is easy to see that for any M > 0 there are primes p such that the numbers 1, 2, …, M are all residues modulo p.

For example, if p ≡ 1 (mod 8), (mod 12), (mod 5) and (mod 28), then by the law of quadratic reciprocity 2, 3, 5, and 7 will all be residues modulo p, and thus all numbers 1–10 will be. The CRT says that this is the same as p ≡ 1 (mod 840), and Dirichlet's theorem says there are an infinite number of primes of this form. 2521 is the smallest, and indeed 1² ≡ 1, 1046² ≡ 2, 123² ≡ 3, 2² ≡ 4, 643² ≡ 5, 87² ≡ 6, 668² ≡ 7, 429² ≡ 8, 3² ≡ 9, and 529² ≡ 10 (mod 2521).

Dirichlet's formulas

The first of these regularities stems from Peter Gustav Lejeune Dirichlet's work (in the 1830s) on the analytic formula for the class number of binary quadratic forms.^[17] Let q be a prime number, s a complex variable, and define a Dirichlet L-function as

L (s) = \sum_{n = 1}^{\infty} (\frac{n}{q}) n^{- s} .

Dirichlet showed that if q ≡ 3 (mod 4), then

L (1) = - \frac{π}{\sqrt{q}} \sum_{n = 1}^{q - 1} \frac{n}{q} (\frac{n}{q}) > 0 .

Therefore, in this case (prime q ≡ 3 (mod 4)), the sum of the quadratic residues minus the sum of the nonresidues in the range 1, 2, …, q − 1 is a negative number.

For example, modulo 11,

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (residues in bold)

1 + 4 + 9 + 5 + 3 = 22, 2 + 6 + 7 + 8 + 10 = 33, and the difference is −11.

In fact the difference will always be an odd multiple of q if q > 3.^[18] In contrast, for prime q ≡ 1 (mod 4), the sum of the quadratic residues minus the sum of the nonresidues in the range 1, 2, …, q − 1 is zero, implying that both sums equal $\frac{q (q - 1)}{4}$ .

Dirichlet also proved that for prime q ≡ 3 (mod 4),

L (1) = \frac{π}{(2 - (\frac{2}{q})) \sqrt{q}} \sum_{n = 1}^{\frac{q - 1}{2}} (\frac{n}{q}) > 0 .

This implies that there are more quadratic residues than nonresidues among the numbers 1, 2, …, (q − 1)/2.

For example, modulo 11 there are four residues less than 6 (namely 1, 3, 4, and 5), but only one nonresidue (2).

An intriguing fact about these two theorems is that all known proofs rely on analysis; no-one has ever published a simple or direct proof of either statement.^[19]

Pairs of residues and nonresidues

Modulo a prime p, the number of pairs n, n + 1 where n R p and n + 1 R p, or n N p and n + 1 R p, etc., are almost equal. More precisely,^[20]^[21]

Let p be an odd prime. For i, j = 0, 1 define the sets

A_{i j} = {k \in {1, 2, \dots, p - 2} : (\frac{k}{p}) = (- 1)^{i} \land (\frac{k + 1}{p}) = (- 1)^{j}},

and let

α_{i j} = | A_{i j} | .

That is,

α₀₀ is the number of residues that are followed by a residue,

α₀₁ is the number of residues that are followed by a nonresidue,

α₁₀ is the number of nonresidues that are followed by a residue, and

α₁₁ is the number of nonresidues that are followed by a nonresidue.

Then if p ≡ 1 (mod 4)

α_{00} = \frac{p - 5}{4}, α_{01} = α_{10} = α_{11} = \frac{p - 1}{4}

and if p ≡ 3 (mod 4)

α_{01} = \frac{p + 1}{4}, α_{00} = α_{10} = α_{11} = \frac{p - 3}{4} .

For example: (residues in bold)
Modulo 17

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
A₀₀ = {1,8,15},

A₀₁ = {2,4,9,13},

A₁₀ = {3,7,12,14},

A₁₁ = {5,6,10,11}.

Modulo 19

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
A₀₀ = {4,5,6,16},

A₀₁ = {1,7,9,11,17},

A₁₀ = {3,8,10,15},

A₁₁ = {2,12,13,14}.

Gauss (1828)^[22] introduced this sort of counting when he proved that if p ≡ 1 (mod 4) then x⁴ ≡ 2 (mod p) can be solved if and only if p = a² + 64 b².

The Pólya–Vinogradov inequality

The values of $(\frac{a}{p})$ for consecutive values of a mimic a random variable like a coin flip.^[23] Specifically,
Pólya and Vinogradov proved^[24] (independently) in 1918 that for any nonprincipal Dirichlet character χ(n) modulo q and any integers M and N,

| \sum_{n = M + 1}^{M + N} χ (n) | = O (\sqrt{q} \log q),

in big O notation. Setting

χ (n) = (\frac{n}{q}),

this shows that the number of quadratic residues modulo q in any interval of length N is

\frac{1}{2} N + O (\sqrt{q} \log q) .

It is easy^[25] to prove that

| \sum_{n = M + 1}^{M + N} (\frac{n}{q}) | < \sqrt{q} \log q .

In fact,^[26] $| \sum_{n = M + 1}^{M + N} (\frac{n}{q}) | < \frac{4}{π^{2}} \sqrt{q} \log q + 0.41 \sqrt{q} + 0.61 .$

Montgomery and Vaughan improved this in 1977, showing that, if the generalized Riemann hypothesis is true then

| \sum_{n = M + 1}^{M + N} χ (n) | = O (\sqrt{q} \log \log q) .

This result cannot be substantially improved, for Schur had proved in 1918 that

\max_{N} | \sum_{n = 1}^{N} (\frac{n}{q}) | > \frac{1}{2 π} \sqrt{q}

and Paley had proved in 1932 that

\max_{N} | \sum_{n = 1}^{N} (\frac{d}{n}) | > \frac{1}{7} \sqrt{d} \log \log d

for infinitely many d > 0.

Least quadratic non-residue

The least quadratic residue mod p is clearly 1. The question of the magnitude of the least quadratic non-residue n(p) is more subtle. The Pólya–Vinogradov inequality above gives O(√p log p). The best unconditional estimate is n(p) ≪ p^θ for any θ>1/4√e, obtained by estimates of Burgess on character sums.^[27] On the assumption of the Generalised Riemann hypothesis, Ankeny obtained n(p) ≪ (log p)².^[28] Linnik has shown that the number of p less than X such that n(p) > X^ε is bounded by a constant depending on ε.^[27]

Quadratic excess

Let p be an odd prime. The quadratic excess E(p) is the number of quadratic residues on the range (0,p/2) minus the number in the range (p/2,p). For p congruent to 1 mod 4, the excess is zero, since −1 is a quadratic residue and the residues are symmetric under r ↔ p−r. For p congruent to 3 mod 4, the excess E is always positive.^[29]

Complexity of finding square roots

That is, given a number a and a modulus n, how hard is it

to tell whether an x solving x² ≡ a (mod n) exists
assuming one does exist, to calculate it?

An important difference between prime and composite moduli shows up here. Modulo a prime p, a quadratic residue a has 1 + (a|p) roots (i.e. zero if a N p, one if a ≡ 0 (mod p), or two if a R p and gcd(a,p) = 1.)

In general if a composite modulus n is written as a product of powers of distinct primes, and there are n₁ roots modulo the first one, n₂ mod the second, …, there will be n₁n₂… roots modulo n.

The theoretical way solutions modulo the prime powers are combined to make solutions modulo n is called the Chinese remainder theorem; it can be implemented with an efficient algorithm.^[30]

For example:
Solve x^{2 ≡ 6 (mod 15).
x² ≡ 6 (mod 3) has one solution, 0; x² ≡ 6 (mod 5) has two, 1 and 4.
and there are two solutions modulo 15, namely 6 and 9.}

Solve x^{2 ≡ 4 (mod 15).
x² ≡ 4 (mod 3) has two solutions, 1 and 2; x² ≡ 4 (mod 5) has two, 2 and 3.
and there are four solutions modulo 15, namely 2, 7, 8, and 13.}

Prime or prime power modulus

First off, if the modulus n is prime the Legendre symbol (a|n) can be quickly computed using a variation of Euclid's algorithm.;^[31] if it is −1 there is no solution. Secondly, assuming that (a|n) = 1, if n ≡ 3 (mod 4), Lagrange found that the solutions are given by

x \equiv \pm a^{(n + 1) / 4} (\mod n),

and Legendre found a similar solution^[32] if n ≡ 5 (mod 8).

For prime n ≡ 1 (mod 8), however, there is no known formula. Tonelli^[33] (in 1891) and Cipolla^[34] found efficient algorithms that work for all prime moduli. Both algorithms require finding a quadratic nonresidue modulo n, and there is no efficient deterministic algorithm known for doing that. But since half the numbers between 1 and n are nonresidues, picking numbers x at random and calculating the Legendre symbol (x|n) until a nonresidue is found will quickly produce one.

If the modulus n is a prime power n = p^e, a solution may be found modulo p and "lifted" to a solution modulo n using Hensel's lemma or an algorithm of Gauss.^[8]

Composite modulus

If the modulus n has been factored into prime powers the solution was discussed above.

If the Jacobi symbol (a|n) = −1 then there is no solution. If it is +1, there may or may not be one.

If the factorization of n is not known, and (a|n) = 1, the problem is known to be equivalent to integer factorization of n (i.e. an efficient solution to either problem could be used to solve the other efficiently).

The above discussion indicates how knowing the factors of n allows us to find the roots efficiently. Say there were an efficient algorithm for finding square roots modulo a composite number. The article congruence of squares discusses how finding two numbers x and y where x² ≡ y² (mod n) and x ≠ ±y suffices to factorize n efficiently. Generate a random number, square it modulo n, and have the efficient square root algorithm find a root. Repeat until it returns a number not equal to the one we originally squared (or its negative modulo n), then follow the algorithm described in congruence of squares. The efficiency of the factoring algorithm depends on the exact characteristics of the root-finder (e.g. does it return all roots? just the smallest one? a random one?), but it will be efficient.^[35]

Simply determining whether a N n or a R n (which can be done efficiently for prime n by computing the Legendre symbol) is known as the quadratic residuosity problem when n is composite. It is not known to be as hard as factorization, but is thought to be quite hard.

On the other hand, if we want to know if there is a solution for x less than some given limit c, this problem is NP-complete;^[36] however, this is a fixed-parameter tractable problem, where c is the parameter.

In general, to determine if a quadratic congruence with composite modulus is solvable use the following theorem:^[37]

Let n > 1, and gcd(a,n) =1. Then x² ≡ a (mod n) is solvable if and only if:

a) The Legendre symbol, (a/p) = 1 for all odd prime divisors of n.

b) a ≡ 1 (mod 4) if 4|n, but 8 does not divide n; a ≡ 1 (mod 8) if 8|n.

Note: This theorem essentially requires that the factorization of n is known. Also notice that if gcd(a,n)=m, then the congruence can be reduced to a/m ≡ x²/m (mod n/m), but then this takes the problem away from being a problem of quadratic congruences (unless m is a square).

The number of quadratic residues

The list of the number of quadratic residues mod n, for n=1,2,3 ..., looks like:^[38]

1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, 4, 9, 8, 10, 6, 8, 12, 12, 6, 11, 14, 11, 8, 15, 12, 16, 7, 12, 18, 12, 8, 19, 20, 14, 9, 21, 16, 22, 12, 12, 24, 24, 8, 22, 22, 18, 14, 27, 22, 18, 12, 20, 30, 30, 12, 31, 32, 16, 12, 21, ...

A formula to count the number of squares mod n is given by Stangl.^[39]

Applications of quadratic residues

Acoustics

Sound diffusers have been based on number-theoretic concepts such as primitive roots and quadratic residues.^[40]

Graph theory

Paley graphs are dense undirected graphs, one for each prime p ≡ 1 (mod 4), that form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices.

Paley digraphs are directed analogs of Paley graphs, one for each p ≡ 3 (mod 4), that yield antisymmetric conference matrices.

The construction of these graphs uses quadratic residues.

Cryptography

The fact that finding a square root of a number modulo a large composite n is equivalent to factoring (which is widely believed be a hard problem) has been used for constructing cryptographic schemes such as the Rabin cryptosystem and the oblivious transfer. The quadratic residuosity problem is the basis for the Goldwasser-Micali cryptosystem.

The discrete logarithm is a similar problem that is also used in cryptography.

Primality testing

Euler's criterion is a formula for the Legendre symbol (a|p) where p is prime. If p is composite the formula may or may not compute (a|p) correctly. The Solovay-Strassen primality test for whether a given number n is prime or composite picks a random a and computes (a|n) using a modification of Euclid's algorithm,^[41] and also using Euler's criterion.^[42] If the results disagree, n is composite; if they agree, n may be composite or prime. For a composite n at least 1/2 the values of a in the range 2, 3, ..., n − 1 will return "n is composite"; for prime n none will. If, after using many different values of a, n has not been proved composite it is called a "probable prime".

The Miller-Rabin primality test is based on the same principles. There is a deterministic version of it, but the proof that it works depends on the generalized Riemann hypothesis; the output from this test is "n is definitely composite" or "either n is prime or the GRH is false". If the second output ever occurs for a composite n, then the GRH would be false, which would have implications through many branches of mathematics.

Integer factorization

In § VI of the Disquisitiones Arithmeticae^[43] Gauss discusses two factoring algorithms that use quadratic residues and the law of quadratic reciprocity.

Several modern factorization algorithms (including Dixon's algorithm, the continued fraction method, the quadratic sieve, and the number field sieve) generate small quadratic residues (modulo the number being factorized) in an attempt to find a congruence of squares which will yield a factorization. The number field sieve is the fastest general-purpose factorization algorithm known.

Table of quadratic residues

Quadratic Residues
x	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
x²	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225	256	289	324	361	400	441	484	529	576	625
mod 2	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
mod 3	1	1	0	1	1	0	1	1	0	1	1	0	1	1	0	1	1	0	1	1	0	1	1	0	1
mod 4	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
mod 5	1	4	4	1	0	1	4	4	1	0	1	4	4	1	0	1	4	4	1	0	1	4	4	1	0
mod 6	1	4	3	4	1	0	1	4	3	4	1	0	1	4	3	4	1	0	1	4	3	4	1	0	1
mod 7	1	4	2	2	4	1	0	1	4	2	2	4	1	0	1	4	2	2	4	1	0	1	4	2	2
mod 8	1	4	1	0	1	4	1	0	1	4	1	0	1	4	1	0	1	4	1	0	1	4	1	0	1
mod 9	1	4	0	7	7	0	4	1	0	1	4	0	7	7	0	4	1	0	1	4	0	7	7	0	4
mod 10	1	4	9	6	5	6	9	4	1	0	1	4	9	6	5	6	9	4	1	0	1	4	9	6	5
mod 11	1	4	9	5	3	3	5	9	4	1	0	1	4	9	5	3	3	5	9	4	1	0	1	4	9
mod 12	1	4	9	4	1	0	1	4	9	4	1	0	1	4	9	4	1	0	1	4	9	4	1	0	1
mod 13	1	4	9	3	12	10	10	12	3	9	4	1	0	1	4	9	3	12	10	10	12	3	9	4	1
mod 14	1	4	9	2	11	8	7	8	11	2	9	4	1	0	1	4	9	2	11	8	7	8	11	2	9
mod 15	1	4	9	1	10	6	4	4	6	10	1	9	4	1	0	1	4	9	1	10	6	4	4	6	10
mod 16	1	4	9	0	9	4	1	0	1	4	9	0	9	4	1	0	1	4	9	0	9	4	1	0	1
mod 17	1	4	9	16	8	2	15	13	13	15	2	8	16	9	4	1	0	1	4	9	16	8	2	15	13
mod 18	1	4	9	16	7	0	13	10	9	10	13	0	7	16	9	4	1	0	1	4	9	16	7	0	13
mod 19	1	4	9	16	6	17	11	7	5	5	7	11	17	6	16	9	4	1	0	1	4	9	16	6	17
mod 20	1	4	9	16	5	16	9	4	1	0	1	4	9	16	5	16	9	4	1	0	1	4	9	16	5
mod 21	1	4	9	16	4	15	7	1	18	16	16	18	1	7	15	4	16	9	4	1	0	1	4	9	16
mod 22	1	4	9	16	3	14	5	20	15	12	11	12	15	20	5	14	3	16	9	4	1	0	1	4	9
mod 23	1	4	9	16	2	13	3	18	12	8	6	6	8	12	18	3	13	2	16	9	4	1	0	1	4
mod 24	1	4	9	16	1	12	1	16	9	4	1	0	1	4	9	16	1	12	1	16	9	4	1	0	1
mod 25	1	4	9	16	0	11	24	14	6	0	21	19	19	21	0	6	14	24	11	0	16	9	4	1	0

Notes

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References

The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 A7.1: AN1, pg.249.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.

Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.
Template:PlanetMath

ja:平方剰余 zh-classical:二次剩餘

↑ Lemmemeyer, Ch. 1
↑ Lemmermeyer, pp 6–8, p. 16 ff
↑ Gauss, DA, art. 94
↑ ^4.0 ^4.1 Gauss, DA, art. 96
↑ ^5.0 ^5.1 Gauss, DA, art. 98
↑ Gauss, DA, art 111
↑ Gauss, DA, art. 103
↑ ^8.0 ^8.1 Gauss, DA, art. 101
↑ Gauss, DA, art. 102
↑ e.g., Template:Harvnb
↑ Gauss, DA, art. 131
↑ e.g. Hardy and Wright use it
↑ Gauss, DA, art 230 ff.
↑ This extension of the domain is necessary for defining L functions.
↑ See Legendre symbol#Formulas for the Legendre symbol for examples
↑ Lemmermeyer, pp 111–end
↑ Template:Harvnb. These are classical results.
↑ Template:Harvnb, (conjectured by Jacobi, proved by Dirichlet)
↑ Template:Harvnb
↑ Lemmermeyer, p. 29 ex. 1.22; cf pp. 26–27, Ch. 10
↑ Crandall & Pomerance, ex 2.38, pp 106–108
↑ Gauss, Theorie der biquadratischen Reste, Erste Abhandlung (pp 511–533 of the Untersuchungen über hohere Arithmetik)
↑ Crandall & Pomerance, ex 2.38, pp 106–108 discuss the similarities and differences. For example, tossing n coins, it is possible (though unlikely) to get n/2 heads followed by that many tails. V-P inequality rules that out for residues.
↑ Template:Harvnb, (proof of P–V, (in fact big-O can be replace by 2); journal references for Paley, Montgomery, and Schur)
↑ Planet Math: Proof of Pólya–Vinogradov Inequality in external links. The proof is a page long and only requires elementary facts about Gaussian sums
↑ Pomerance & Crandall, ex 2.38 pp.106–108. result from T. Cochrane, "On a trigonometric inequality of Vinogradov", J. Number Theory, 27:9–16, 1987
↑ ^27.0 ^27.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ Template:Harvnb; it requires O(log² m) steps where m is the number of primes dividing n.
↑ Template:Harvnb; computing (a|n) requires O(log a log n) steps
↑ Lemmermeyer, p. 29
↑ Template:Harvnb; the algorithm requires O(log⁴n) steps.
↑ Template:Harvnb; the algorithm requires O(log³ n) steps and is also nondetermisitic.
↑ Crandall & Pomerance, ex. 6.5 & 6.6, p.273
↑ Template:Harvnb
↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ Before you resolve whether or not chrome steel cookware is value buying, lets first discuss what chrome steel cookware is. Chrome steel is made from an alloy, or a mixture of metals. Most commonly, fundamental iron with chromium, nickel or some other minor metals. The chromium gives rust safety and gives your cookware durability. The nickel supplies rust protection as effectively, and provides a polished look. Most properly made stainless steel cookware has copper or aluminum added to the underside of the pan or pot. That is done to increases the power of the pot or pan to conduct heat.
The perfect stainless-steel cookware is the primary category, however still it is divided into several subcategories based mostly on the standard and the price range. It may be complicated to choose the most effective stainless steel cookware out of the classes that can meet your necessities. That is where we took a step forward to explain you all the knowledge that might be useful for you to understand how to choose the most effective stainless-steel cookware. The very best stainless-steel cookware set is manufactured from low-cost to costly and quality constructed pots and pans.
You'll discover magnetic stainless-steel in the layer on the surface of some quality pieces of chrome steel. That is to make it compatible with induction stovetops, which contain using a quickly charging electromagnetic discipline to warmth cookware. Excessive-quality stainless-steel, like All-Clad , uses three layers of metallic—the austenite layer of steel on the inside, ferrite metal on the skin, and a layer of aluminum sandwiched between the 2 for optimal warmth conductivity (steel alone doesn't conduct warmth evenly). Lesser-high quality stainless steel is normally just one layer of austenitic chrome steel.
Aesthetically speaking, chrome steel is a sensible alternative should you want to display or hang pots or pans. The clear, crisp look of all stainless-steel kitchenware can transform a mishmash of cookware into a complicated décor statement. In case you adored this informative article as well as you wish to receive details relating to best stainless steel cookware i implore you to go to the page. Chrome steel kettles, such as the Cuisinart Tea Kettle will mix particular person kitchenware right into a cohesive and pleasant entity. Contemplate buying stainless steel utensils as nicely. Already acquired a stunning stainless-steel cookware assortment? The Cuisinart Chef’s Assortment stainless pot rack is perhaps the completion for a kitchen, releasing up house and making these pots and pans readily accessible. Get the stainless-steel cookware of your culinary dreams at Macy’s!
Hard-anodized aluminum cookware is one of the most popular forms of material, despite the fact that many individuals do not quite perceive the construction. Onerous-anodized aluminum is plain aluminum that has been processed in a series of chemical baths charged with an electrical current. The result is a cloth that has the same superior warmth conductivity as aluminum however is non-reactive with acidic meals, reminiscent of tomatoes, and twice as laborious as chrome steel. Two drawbacks to onerous-anodized cookware are that it is not dishwasher-safe and, as a result of it is not magnetic, it will not work with induction vary tops.
The enamel over steel approach creates a chunk that has the heat distribution of carbon metal and a non-reactive, low-stick floor. Such pots are a lot lighter than most different pots of comparable dimension, are cheaper to make than stainless steel pots, and don't have the rust and reactivity problems with forged iron or carbon steel. quotation needed Enamel over steel is right for large stockpots and for other massive pans used principally for water-primarily based cooking. Due to its light weight and easy cleanup, enamel over steel is also popular for cookware used while camping. Clad aluminium or copper edit
Unique specialty cookware pieces served a la carte to compliment any cookware set are constructed of a sturdy Stainless Steel with a brushed exterior finish. Designed with an affect bonded, aluminum disk encapsulated base which distributes warmth shortly and evenly to allow precise temperature management. Handles are riveted for sturdiness and efficiency. The New Specialty Cookware is appropriate for all vary sorts together with induction. In addition to the multi use operate, another unique function is bottom to high inside quantity markings in both quarts and metric measurement; and each piece comes with a tempered glass lid, oven protected to 350°F.
Whether you're a cooking fans, an expert chef or just cooking for your family you recognize the significance of getting a totally stocked kitchen. Not only do you want the appropriate components, however you additionally want the proper instruments to get the job performed. In any type of primary cooking training lesson, you'll study that chrome steel is your new greatest good friend with regards to kitchen cookware. What you will also be taught is that quality cooking gear does not normally come at a reduced worth. For that reason, it is important to take excellent care of your cookware! Listed below are some fundamentals for stainless steel care.
To fight the uneven heating drawback, most stainless-steel pans are laminations of aluminum or copper on the underside to unfold the warmth round, and stainless-steel contained in the pan to supply a cooking floor that's impervious to whatever you may put inside. In my experience, this chrome steel surface is still too sticky to fry on, and for those who ever burn it you get a everlasting trouble spot. But, sometimes a stainless steel cooking surface comes in handy when you can't use aluminum (see below) so I hold some around. Select something with a fairly thick aluminum layer on the bottom.
Effectively, unless you’re a metals expert and go examine the manufacturing unit the place the metal is made to see whether or not their manufacturing course of creates a pure austenite with out corrosive supplies shaped, you’re not going to know for certain whether or not the craftsmanship of your stainless is of the very best high quality. I feel your greatest bet is to simply purchase excessive-quality stainless steel from the beginning, from a model with a status for good quality. But, I think I've figured out a technique that you could determine if the stainless cookware you have already got is potentially reactive.
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ Template:Cite web
↑ Template:Harvnb
↑ Template:Harvnb; Euler's criterion requires O(log³ n) steps
↑ Gauss, DA, arts 329–334

[1] Lemmemeyer, Ch. 1

[2] Lemmermeyer, pp 6–8, p. 16 ff

[3] Gauss, DA, art. 94

[Gauss,_DA,_art._96-4] 4.0 ^4.1 Gauss, DA, art. 96

[Gauss,_DA,_art._98-5] 5.0 ^5.1 Gauss, DA, art. 98

[6] Gauss, DA, art 111

[7] Gauss, DA, art. 103

[Gauss,_DA,_art._101-8] 8.0 ^8.1 Gauss, DA, art. 101

[9] Gauss, DA, art. 102

[10] .g., Template:Harvnb

[11] Gauss, DA, art. 131

[12] .g. Hardy and Wright use it

[13] Gauss, DA, art 230 ff.

[14] This extension of the domain is necessary for defining L functions.

[15] See Legendre symbol#Formulas for the Legendre symbol for examples

[16] Lemmermeyer, pp 111–end

[17] Template:Harvnb. These are classical results.

[18] Template:Harvnb, (conjectured by Jacobi, proved by Dirichlet)

[19] Template:Harvnb

[20] Lemmermeyer, p. 29 ex. 1.22; cf pp. 26–27, Ch. 10

[21] Crandall & Pomerance, ex 2.38, pp 106–108

[22] Gauss, Theorie der biquadratischen Reste, Erste Abhandlung (pp 511–533 of the Untersuchungen über hohere Arithmetik)

[23] Crandall & Pomerance, ex 2.38, pp 106–108 discuss the similarities and differences. For example, tossing n coins, it is possible (though unlikely) to get n/2 heads followed by that many tails. V-P inequality rules that out for residues.

[24] Template:Harvnb, (proof of P–V, (in fact big-O can be replace by 2); journal references for Paley, Montgomery, and Schur)

[25] Planet Math: Proof of Pólya–Vinogradov Inequality in external links. The proof is a page long and only requires elementary facts about Gaussian sums

[26] Pomerance & Crandall, ex 2.38 pp.106–108. result from T. Cochrane, "On a trigonometric inequality of Vinogradov", J. Number Theory, 27:9–16, 1987

[FI156-27] 27.0 ^27.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[28] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[29] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[30] Template:Harvnb; it requires O(log² m) steps where m is the number of primes dividing n.

[31] Template:Harvnb; computing (a|n) requires O(log a log n) steps

[32] Lemmermeyer, p. 29

[33] Template:Harvnb; the algorithm requires O(log⁴n) steps.

[34] Template:Harvnb; the algorithm requires O(log³ n) steps and is also nondetermisitic.

[35] Crandall & Pomerance, ex. 6.5 & 6.6, p.273

[36] Template:Harvnb

[37] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[38] Before you resolve whether or not chrome steel cookware is value buying, lets first discuss what chrome steel cookware is. Chrome steel is made from an alloy, or a mixture of metals. Most commonly, fundamental iron with chromium, nickel or some other minor metals. The chromium gives rust safety and gives your cookware durability. The nickel supplies rust protection as effectively, and provides a polished look. Most properly made stainless steel cookware has copper or aluminum added to the underside of the pan or pot. That is done to increases the power of the pot or pan to conduct heat.
The perfect stainless-steel cookware is the primary category, however still it is divided into several subcategories based mostly on the standard and the price range. It may be complicated to choose the most effective stainless steel cookware out of the classes that can meet your necessities. That is where we took a step forward to explain you all the knowledge that might be useful for you to understand how to choose the most effective stainless-steel cookware. The very best stainless-steel cookware set is manufactured from low-cost to costly and quality constructed pots and pans.
You'll discover magnetic stainless-steel in the layer on the surface of some quality pieces of chrome steel. That is to make it compatible with induction stovetops, which contain using a quickly charging electromagnetic discipline to warmth cookware. Excessive-quality stainless-steel, like All-Clad , uses three layers of metallic—the austenite layer of steel on the inside, ferrite metal on the skin, and a layer of aluminum sandwiched between the 2 for optimal warmth conductivity (steel alone doesn't conduct warmth evenly). Lesser-high quality stainless steel is normally just one layer of austenitic chrome steel.
Aesthetically speaking, chrome steel is a sensible alternative should you want to display or hang pots or pans. The clear, crisp look of all stainless-steel kitchenware can transform a mishmash of cookware into a complicated décor statement. In case you adored this informative article as well as you wish to receive details relating to best stainless steel cookware i implore you to go to the page. Chrome steel kettles, such as the Cuisinart Tea Kettle will mix particular person kitchenware right into a cohesive and pleasant entity. Contemplate buying stainless steel utensils as nicely. Already acquired a stunning stainless-steel cookware assortment? The Cuisinart Chef’s Assortment stainless pot rack is perhaps the completion for a kitchen, releasing up house and making these pots and pans readily accessible. Get the stainless-steel cookware of your culinary dreams at Macy’s!
Hard-anodized aluminum cookware is one of the most popular forms of material, despite the fact that many individuals do not quite perceive the construction. Onerous-anodized aluminum is plain aluminum that has been processed in a series of chemical baths charged with an electrical current. The result is a cloth that has the same superior warmth conductivity as aluminum however is non-reactive with acidic meals, reminiscent of tomatoes, and twice as laborious as chrome steel. Two drawbacks to onerous-anodized cookware are that it is not dishwasher-safe and, as a result of it is not magnetic, it will not work with induction vary tops.
The enamel over steel approach creates a chunk that has the heat distribution of carbon metal and a non-reactive, low-stick floor. Such pots are a lot lighter than most different pots of comparable dimension, are cheaper to make than stainless steel pots, and don't have the rust and reactivity problems with forged iron or carbon steel. quotation needed Enamel over steel is right for large stockpots and for other massive pans used principally for water-primarily based cooking. Due to its light weight and easy cleanup, enamel over steel is also popular for cookware used while camping. Clad aluminium or copper edit
Unique specialty cookware pieces served a la carte to compliment any cookware set are constructed of a sturdy Stainless Steel with a brushed exterior finish. Designed with an affect bonded, aluminum disk encapsulated base which distributes warmth shortly and evenly to allow precise temperature management. Handles are riveted for sturdiness and efficiency. The New Specialty Cookware is appropriate for all vary sorts together with induction. In addition to the multi use operate, another unique function is bottom to high inside quantity markings in both quarts and metric measurement; and each piece comes with a tempered glass lid, oven protected to 350°F.
Whether you're a cooking fans, an expert chef or just cooking for your family you recognize the significance of getting a totally stocked kitchen. Not only do you want the appropriate components, however you additionally want the proper instruments to get the job performed. In any type of primary cooking training lesson, you'll study that chrome steel is your new greatest good friend with regards to kitchen cookware. What you will also be taught is that quality cooking gear does not normally come at a reduced worth. For that reason, it is important to take excellent care of your cookware! Listed below are some fundamentals for stainless steel care.
To fight the uneven heating drawback, most stainless-steel pans are laminations of aluminum or copper on the underside to unfold the warmth round, and stainless-steel contained in the pan to supply a cooking floor that's impervious to whatever you may put inside. In my experience, this chrome steel surface is still too sticky to fry on, and for those who ever burn it you get a everlasting trouble spot. But, sometimes a stainless steel cooking surface comes in handy when you can't use aluminum (see below) so I hold some around. Select something with a fairly thick aluminum layer on the bottom.
Effectively, unless you’re a metals expert and go examine the manufacturing unit the place the metal is made to see whether or not their manufacturing course of creates a pure austenite with out corrosive supplies shaped, you’re not going to know for certain whether or not the craftsmanship of your stainless is of the very best high quality. I feel your greatest bet is to simply purchase excessive-quality stainless steel from the beginning, from a model with a status for good quality. But, I think I've figured out a technique that you could determine if the stainless cookware you have already got is potentially reactive.

[39] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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[42] Template:Harvnb; Euler's criterion requires O(log³ n) steps

[43] Gauss, DA, arts 329–334

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Angular frequency: Difference between revisions

Revision as of 18:59, 11 November 2013

Contents

History, conventions, and elementary facts

Prime modulus

Prime power modulus

Composite modulus not a prime power

Notations

Distribution of quadratic residues

Dirichlet's formulas

Pairs of residues and nonresidues

The Pólya–Vinogradov inequality

Least quadratic non-residue

Quadratic excess

Complexity of finding square roots

Prime or prime power modulus

Composite modulus

The number of quadratic residues

Applications of quadratic residues

Acoustics

Graph theory

Cryptography

Primality testing

Integer factorization

Table of quadratic residues

See also

Notes

References

External links

Navigation menu

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+In [[number theory]], an [[integer]] ''q'' is called a '''quadratic residue''' [[modular arithmetic|modulo]] ''n'' if it is [[Congruence relation|congruent]] to a [[Square number|perfect square]] modulo ''n''; i.e., if there exists an integer ''x'' such that:
+:<math>x^2\equiv q \pmod{n}.</math>
+Otherwise, ''q'' is called a  '''quadratic nonresidue''' modulo ''n''.
+Originally an abstract [[mathematics|mathematical]] concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to [[cryptography]] and the [[Integer factorization|factoring of large numbers]].
+==History, conventions, and elementary facts==
+[[Fermat]], [[Euler]], [[Joseph Louis Lagrange|Lagrange]], [[Adrien-Marie Legendre|Legendre]], and other number theorists of the 17th and 18th centuries proved some theorems<ref>Lemmemeyer, Ch. 1</ref> and made some conjectures<ref>Lemmermeyer, pp 6&ndash;8, p. 16 ff</ref> about quadratic residues, but the first systematic treatment is § IV of [[Gauss|Gauss's]] ''[[Disquisitiones Arithmeticae]]'' (1801). Article 95 introduces the terminology "quadratic residue" and "quadratic nonresidue", and states that, if the context makes it clear, the adjective "quadratic" may be dropped.
+For a given ''n'' a list of the quadratic residues modulo ''n'' may be obtained by simply squaring the numbers 0, 1, …, ''n'' &minus; 1. Because ''a''<sup>2</sup> ≡ (''n'' &minus; ''a'')<sup>2</sup> (mod ''n''), the list of squares modulo ''n'' is symmetrical around ''n''/2, and the list only needs to go that high. This can be seen in the table [[#Table of quadratic residues|below]].
+Thus, the number of quadratic residues modulo ''n'' cannot exceed ''n''/2 + 1 (''n'' even) or (''n'' + 1)/2 (''n'' odd).<ref>Gauss, DA, art. 94</ref>
+The product of two residues is always a residue.
+===Prime modulus===
+Modulo 2, every integer is a quadratic residue.
+Modulo an odd [[prime number]] ''p'' there are (''p'' + 1)/2 residues (including 0) and (''p'' &minus; 1)/2 nonresidues. In this case, it is customary to consider 0 as a special case and work within the  [[Multiplicative group of integers modulo n|multiplicative group of nonzero elements]] of the [[Field (mathematics)|field]] '''Z/''p''Z'''. (In other words, every congruence class except zero modulo ''p'' has a multiplicative inverse. This is not true for composite moduli.)<ref name="Gauss, DA, art. 96">Gauss, DA, art. 96</ref>
+Following this convention, the multiplicative inverse of a residue is a residue, and the inverse of a nonresidue is a nonresidue.<ref name="Gauss, DA, art. 98">Gauss, DA, art. 98</ref>
+Following this convention, modulo a prime number there are an equal number of residues and nonresidues.<ref name="Gauss, DA, art. 96"/>
+Modulo a prime, the product of two nonresidues is a residue and the product of a nonresidue and a (nonzero) residue is a nonresidue.<ref name="Gauss, DA, art. 98"/>
+The first supplement<ref>Gauss, DA, art 111</ref> to the [[law of quadratic reciprocity]] is that if ''p'' ≡ 1 (mod 4) then &minus;1 is a quadratic residue modulo ''p'', and if ''p'' ≡ 3 (mod 4) then &minus;1 is a nonresidue modulo ''p''. This implies the following:
+If ''p'' ≡ 1 (mod 4) the negative of a residue modulo ''p'' is a residue and the negative of a nonresidue is a nonresidue.
+If ''p'' ≡ 3 (mod 4) the negative of a residue modulo ''p'' is a nonresidue and the negative of a nonresidue is a residue.
+===Prime power modulus===
+All odd squares are ≡ 1 (mod 8) and ''a fortiori'' ≡ 1 (mod 4). If ''a'' is an odd  number and ''m'' = 8, 16, or some higher power of 2, then ''a'' is a residue modulo ''m'' if and only if ''a'' ≡ 1 (mod 8).<ref>Gauss, DA, art. 103</ref>
+<blockquote>For example, mod (32) the odd squares are
+:1<sup>2</sup> &equiv; 15<sup>2</sup> &equiv; 1
+:3<sup>2</sup> &equiv; 13<sup>2</sup> &equiv; 9
+:5<sup>2</sup> &equiv; 11<sup>2</sup> &equiv; 25
+:7<sup>2</sup> &equiv; 9<sup>2</sup> &equiv; 17
+and the even ones are
+:0<sup>2</sup> &equiv; 8<sup>2</sup> &equiv; 16<sup>2</sup> &equiv; 0
+:2<sup>2</sup> &equiv; 6<sup>2</sup>&equiv; 10<sup>2</sup> &equiv; 14<sup>2</sup>&equiv; 4
+:4<sup>2</sup> &equiv; 12<sup>2</sup> &equiv; 16.
+</blockquote>
+So a nonzero number is a residue mod 8, 16, etc., if and only if it is of the form 4<sup>''k''</sup>(8''n'' + 1).
+A number ''A'' relatively prime to an odd prime ''p'' is a residue modulo any power of ''p'' if and only if it is a residue modulo ''p''.<ref name="Gauss, DA, art. 101">Gauss, DA, art. 101</ref>
+If the modulus is ''p''<sup>''n''</sup>,
+:then ''p''<sup>''k''</sup>''A''
+::is a residue modulo ''p''<sup>''n''</sup> if ''k'' &ge; ''n''
+::is a nonresidue modulo ''p''<sup>''n''</sup> if ''k'' &lt; ''n'' is odd
+::is a residue modulo ''p''<sup>''n''</sup> if ''k'' &lt; ''n'' is even and ''A'' is a residue
+::is a nonresidue modulo ''p''<sup>''n''</sup> if ''k'' &lt; ''n'' is even and ''A'' is a nonresidue.<ref>Gauss, DA, art. 102</ref>
+Notice that the rules are different for powers of two and powers of odd primes.
+Modulo an odd prime power ''n'' = ''p''<sup>''k''</sup>, the products of residues and nonresidues relatively prime to ''p'' obey the same rules as they do mod ''p''; ''p'' is a nonresidue, and in general all the residues and nonresidues obey the same rules, except that the products will be zero if the power of ''p'' in the product ≥ ''n''.
+Modulo 8, the product of the nonresidues 3 and 5 is the nonresidue 7, and likewise for permutations of 3, 5 and 7. In fact, the multiplicative group of the non-residues and 1 form the [[Klein four-group]].
+===Composite modulus not a prime power===
+The basic fact in this case is
+:if ''a'' is a residue modulo ''n'', then ''a'' is a residue modulo ''p''<sup>''k''</sup> for ''every'' prime power dividing ''n''.
+:if ''a'' is a nonresidue modulo ''n'', then ''a'' is a nonresidue modulo ''p''<sup>''k''</sup> for ''at least one'' prime power dividing ''n''.
+Modulo a composite number, the product of two residues is a residue. The product of a residue and a nonresidue may be a residue, a nonresidue, or zero.
+<blockquote>
+For example, from the table for modulus 6 &nbsp;
+ '''1''', 2, '''3''', '''4''', 5 (residues in '''bold''').
+The product of the residue 3 and the nonresidue 5 is the residue 3, whereas the product of the residue 4 and the nonresidue 2 is the nonresidue 2.
+</blockquote>
+Also, the product of two nonresidues may be either a residue, a nonresidue, or zero.
+<blockquote>
+For example, from the table for modulus 15 &nbsp;
+ '''1''', 2, 3, '''4''', 5, '''6''', 7, 8,  '''9''', '''10''', 11, 12, 13, 14 (residues in '''bold''').
+The product of the nonresidues 2 and 8 is the residue 1, whereas the product of the nonresidues 2 and 7 is the nonresidue 14.
+</blockquote>
+This phenomenon can best be described using the vocabulary of abstract algebra. The congruence classes relatively prime to the modulus are a [[Group (mathematics)|group]] under multiplication, called the [[group of units]] of the [[Ring (mathematics)|ring]] '''Z/''n''Z''', and the squares are a [[subgroup]] of it. Different nonresidues may belong to different [[coset]]s, and there is no simple rule that  predicts which one their product will be in. Modulo a prime, there is only the subgroup of squares and a single coset.
+The fact that, e.g., modulo 15 the product of the nonresidues 3 and 5, or of the nonresidue 5 and the residue 9, or the two residues 9 and 10 are all zero comes from working in the full ring '''Z/''n''Z''', which has [[zero divisor]]s for composite ''n''.
+For this reason some authors<ref>e.g., {{harvnb|Ireland|Rosen|1990|p=50}}</ref> add to the definition that a quadratic residue ''q'' must not only be a square but must also be [[relatively prime]] to the modulus ''n''.
+Although it makes things tidier, this article does not insist that residues must be coprime to the modulus.
+==Notations==
+Gauss<ref>Gauss, DA, art. 131</ref> used '''R''' and '''N''' to denote residuacity and non-residuacity, respectively;
+:for example, 2 R 7 and 5 N 7, or 1 R 8 and 3,5,7 N 8.
+Although this notation is compact and convenient for some purposes,<ref>e.g. Hardy and Wright use it</ref><ref>Gauss, DA, art 230 ff.</ref> a more useful notation is the [[Legendre symbol]], also called the [[Dirichlet character#Examples|quadratic character]], which is defined for all integers ''a'' and positive odd [[prime numbers]] ''p'' as
+:<math>
+\left(\frac{a}{p}\right) = \begin{cases}\;\;\,0\mbox{ if }p \mbox { divides } a\\+1\mbox{ if }a \mbox{ R } p \mbox{ and }p \mbox { does not divide } a\\-1\mbox{ if }a \mbox{ N  } p .\end{cases}</math>
+There are two reasons why numbers ≡ 0 (mod ''p'') are treated specially. As we have seen,  it makes many formulas and theorems easier to state. The other (related) reason is that the quadratic character is a [[homomorphism]] from the [[multiplicative group of integers modulo n|multiplicative group of nonzero conguence classes modulo ''p'']] to the [[complex numbers]]. Setting <math>(\tfrac{np}{p}) = 0</math>  allows its [[Domain (mathematics)|domain]] to be extended to the multiplicative [[semigroup]] of all the integers.<ref>This extension of the domain is necessary for defining ''L'' functions.</ref>
+One advantage of this notation over Gauss's is that the Legendre symbol is a function that can be used in formulas.<ref>See [[Legendre symbol#Formulas for the Legendre symbol]] for examples</ref>  It can also easily be generalized to [[Cubic reciprocity|cubic]], quartic and higher power residues.<ref>Lemmermeyer, pp 111&ndash;end</ref>
+There is a generalization of the Legendre symbol for composite values of ''p'', the [[Jacobi symbol]], but its properties are not as simple: if ''m'' is composite and the Jacobi symbol <math>(\tfrac{a}{m}) = -1,</math> then ''a''&nbsp;N&nbsp;''m'', and if ''a''&nbsp;R&nbsp;''m'' then <math>(\tfrac{a}{m}) = 1,</math> but if <math>(\tfrac{a}{m}) = 1</math> we do not know whether ''a''&nbsp;R&nbsp;''m'' or ''a''&nbsp;N&nbsp;''m''. If ''m'' is prime, the Jacobi and Legendre symbols agree.
+==Distribution of quadratic residues==
+Although quadratic residues appear to occur in a rather random pattern modulo ''n'', and this has been exploited in such [[#Applications of quadratic residues|applications]] as [[#Acoustics|acoustics]] and [[#Cryptography|cryptography]], their distribution also exhibits some striking regularities.
+Using [[Dirichlet's theorem on arithmetic progressions|Dirichlet's theorem]] on primes in arithmetic progressions, the [[law of quadratic reciprocity]], and the [[Chinese remainder theorem]] (CRT) it is easy to see that for any ''M'' > 0 there are primes ''p'' such that the numbers 1, 2, …, ''M'' are all residues modulo ''p''.
+<blockquote>For example, if ''p'' ≡ 1 (mod 8), (mod 12), (mod 5) and (mod 28), then by the law of quadratic reciprocity 2, 3, 5, and 7 will all be residues modulo ''p'', and thus all numbers 1&ndash;10 will be. The CRT says that this is the same as ''p'' ≡ 1 (mod 840), and Dirichlet's theorem says there are an infinite number of primes of this form. 2521 is the smallest, and indeed 1<sup>2</sup> ≡ 1, 1046<sup>2</sup> ≡ 2, 123<sup>2</sup> ≡ 3, 2<sup>2</sup> ≡ 4, 643<sup>2</sup> ≡ 5, 87<sup>2</sup> ≡ 6, 668<sup>2</sup> ≡ 7, 429<sup>2</sup> ≡ 8, 3<sup>2</sup> ≡ 9, and  529<sup>2</sup> ≡ 10 (mod 2521).</blockquote>
+===Dirichlet's formulas===
+The first of these regularities stems from [[Peter Gustav Lejeune Dirichlet]]'s work (in the 1830s) on the [[class number formula|analytic formula]] for the [[Class number (number theory)|class number]] of binary [[quadratic form]]s.<ref>{{harvnb|Davenport|2000|pp=8&ndash;9, 43&ndash;51}}. These are classical results.</ref> Let ''q'' be a prime number, ''s'' a complex variable, and define a [[Dirichlet L-function]] as
+:<math>L(s) = \sum_{n=1}^\infty\left(\frac{n}{q}\right)n^{-s}. </math>
+Dirichlet showed that if ''q'' ≡ 3 (mod 4), then
+:<math>L(1) = -\frac{\pi}{\sqrt q}\sum_{n=1}^{q-1} \frac{n}{q} \left(\frac{n}{q}\right) > 0.</math>
+Therefore, in this case (prime ''q'' ≡ 3 (mod 4)), the sum of the quadratic residues minus the sum of the nonresidues in the range 1, 2, …, ''q'' &minus; 1 is a negative number.
+<blockquote>
+For example, modulo 11,
+:'''1''', 2, '''3''', '''4''', '''5''', 6, 7, 8, '''9''', 10 (residues in '''bold''')
+:1 + 4 + 9 + 5 + 3 = 22, 2 + 6 + 7 + 8 + 10 = 33, and the difference is &minus;11. </blockquote>
+In fact the difference will always be an odd multiple of ''q'' if ''q'' > 3.<ref>{{harvnb|Davenport|2000|pp=49&ndash;51}}, (conjectured by [[Carl Gustav Jacob Jacobi|Jacobi]], proved by Dirichlet)</ref> In contrast, for prime ''q'' ≡ 1 (mod 4), the sum of the quadratic residues minus the sum of the nonresidues in the range 1, 2, …, ''q'' &minus; 1 is zero, implying that both sums equal <math>\frac{q(q-1)}{4}</math>.
+Dirichlet also proved that for prime ''q'' ≡ 3 (mod 4),
+:<math>L(1) = \frac{\pi}{\left(2-\left(\frac{2}{q}\right)\right)\!\sqrt q}\sum_{n=1}^\frac{q-1}{2}\left(\frac{n}{q}\right) > 0.</math>
+This implies that there are more quadratic residues than nonresidues among the numbers 1, 2, …, (''q'' &minus; 1)/2.
+<blockquote>For example, modulo 11 there are four residues less than 6 (namely 1, 3, 4, and 5), but only one nonresidue (2).</blockquote>
+An intriguing fact about these two theorems is that all known proofs rely on analysis; no-one has ever published a simple or direct proof of either statement.<ref>{{harvnb|Davenport|2000|p=9}}</ref>
+===Pairs of residues and nonresidues===
+Modulo a prime ''p'', the number of pairs ''n'', ''n'' + 1 where ''n'' R ''p'' and ''n'' + 1 R ''p'', or ''n'' N ''p'' and ''n'' + 1 R ''p'', etc., are almost equal. More precisely,<ref>Lemmermeyer, p. 29 ex. 1.22; cf pp. 26&ndash;27, Ch. 10</ref><ref>Crandall & Pomerance, ex 2.38, pp 106&ndash;108</ref>
+Let ''p'' be an odd prime. For ''i'', ''j'' = 0, 1 define the sets
+:<math>A_{ij}=\left\{k\in\{1,2,\dots,p-2\}: \left(\frac{k}{p}\right)=(-1)^i\land\left(\frac{k+1}{p}\right)=(-1)^j\right\},</math>
+and let
+:<math>\alpha_{ij} = |A_{ij}|.</math>
+That is,
+:&alpha;<sub>00</sub> is the number of residues that are followed by a residue,
+:&alpha;<sub>01</sub> is the number of residues that are followed by a nonresidue,
+:&alpha;<sub>10</sub> is the number of nonresidues that are followed by a residue, and
+:&alpha;<sub>11</sub> is the number of nonresidues that are followed by a nonresidue.
+Then if ''p'' ≡ 1 (mod 4)
+:<math>\alpha_{00} = \frac{p-5}{4},\;\alpha_{01} =\alpha_{10} =\alpha_{11} = \frac{p-1}{4} </math>
+and if ''p'' ≡ 3 (mod 4)
+:<math>\alpha_{01} = \frac{p+1}{4},\;\alpha_{00} =\alpha_{10} =\alpha_{11} = \frac{p-3}{4}. </math>
+<blockquote>For example: (residues in '''bold''')
+Modulo 17
+:'''1''', '''2''', 3, '''4''', 5, 6, 7, '''8''', '''9''', 10, 11, 12, '''13''', 14, '''15''', '''16'''
+::''A''<sub>00</sub> = {1,8,15},
+::''A''<sub>01</sub> = {2,4,9,13},
+::''A''<sub>10</sub> = {3,7,12,14},
+::''A''<sub>11</sub> = {5,6,10,11}.
+Modulo 19
+:'''1''', 2, 3, '''4''', '''5''', '''6''', '''7''', 8, '''9''', 10, '''11''', 12, 13, 14, 15, '''16''', '''17''', 18
+::''A''<sub>00</sub> = {4,5,6,16},
+::''A''<sub>01</sub> = {1,7,9,11,17},
+::''A''<sub>10</sub> = {3,8,10,15},
+::''A''<sub>11</sub> = {2,12,13,14}. </blockquote>
+Gauss (1828)<ref>Gauss, ''Theorie der biquadratischen Reste, Erste Abhandlung'' (pp 511&ndash;533 of the ''Untersuchungen über hohere Arithmetik)''</ref> introduced this sort of counting when he proved that if ''p'' ≡ 1 (mod 4) then ''x''<sup>4</sup> ≡ 2 (mod ''p'') can be solved if and only if ''p''&nbsp;=&nbsp;''a''<sup>2</sup>&nbsp;+&nbsp;64&nbsp;''b''<sup>2</sup>.
+===The Pólya&ndash;Vinogradov inequality===
+The values of <math>(\tfrac{a}{p})</math> for consecutive values of ''a'' mimic a random variable like a [[coin flip]].<ref>Crandall & Pomerance, ex 2.38, pp 106&ndash;108 discuss the similarities and differences. For example, tossing ''n'' coins, it is possible (though unlikely) to get ''n''/2 heads followed by that many tails. V-P inequality rules that out for residues.</ref> Specifically,
+<br>
+[[George Pólya|Pólya]] and [[Ivan Matveevich Vinogradov|Vinogradov]] proved<ref>{{harvnb|Davenport|2000|pp=135–137}}, (proof of P&ndash;V, (in fact big-O can be replace by 2); journal references for Paley, Montgomery, and Schur)</ref> (independently) in 1918 that for any nonprincipal [[Dirichlet character]] χ(''n'') modulo ''q'' and any integers ''M'' and ''N'',
+:<math>\left|\sum_{n=M+1}^{M+N}\chi(n)\right| =O\left( \sqrt q \log q\right),
+</math> in [[big O notation]]. Setting <math> \chi(n) = \left(\frac{n}{q}\right),</math>
+this shows that the number of quadratic residues modulo ''q'' in any interval of length ''N'' is
+:<math>\frac{1}{2}N + O(\sqrt q\log q).</math> It is easy<ref>Planet Math:  Proof of Pólya&ndash;Vinogradov Inequality in [[#External links|external links]]. The proof is  a page long and only requires elementary facts about Gaussian sums</ref> to prove that <math> \left| \sum_{n=M+1}^{M+N} \left( \frac{n}{q} \right) \right| < \sqrt q \log q.</math>
+In fact,<ref>Pomerance & Crandall, ex 2.38 pp.106&ndash;108. result from  T. Cochrane, "On a trigonometric inequality of Vinogradov", ''J. Number Theory'', 27:9&ndash;16, 1987</ref>  <math> \left| \sum_{n=M+1}^{M+N} \left( \frac{n}{q} \right) \right| < \frac{4}{\pi^2} \sqrt q \log q+0.41\sqrt q +0.61.</math>
+[[Hugh Montgomery (mathematician)|Montgomery]] and [[Robert Charles Vaughan (mathematician)|Vaughan]] improved this in 1977, showing that, if the [[generalized Riemann hypothesis]] is true then
+:<math>\left|\sum_{n=M+1}^{M+N}\chi(n)\right|=O\left(\sqrt q \log  \log q\right).</math>
+This result cannot be substantially improved, for [[Issai Schur|Schur]] had proved in 1918 that
+:<math>\max_N \left|\sum_{n=1}^{N}\left(\frac{n}{q}\right)\right|>\frac{1}{2\pi}\sqrt q</math>
+and [[Raymond Paley|Paley]] had proved in 1932 that
+:<math>\max_N \left|\sum_{n=1}^{N}\left(\frac{d}{n}\right)\right|>\frac{1}{7}\sqrt d \log \log d</math>
+for infinitely many ''d'' &gt; 0.
+===Least quadratic non-residue===
+The least quadratic residue mod ''p'' is clearly 1.  The question of the magnitude of the least quadratic non-residue ''n''(''p'') is more subtle.  The Pólya–Vinogradov inequality above gives O(√''p'' log ''p'').  The best unconditional estimate is ''n''(''p'') ≪ ''p''<sup>θ</sup> for any θ>1/4√e, obtained by estimates of Burgess on [[character sum]]s.<ref name=FI156/>  On the assumption of the [[Generalised Riemann hypothesis]], Ankeny obtained ''n''(''p'') ≪ (log ''p'')<sup>2</sup>.<ref>{{cite book | title=Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis | first=Hugh L. | last=Montgomery | authorlink=Hugh Montgomery (mathematician) | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0737-4 | zbl=0814.11001 | page=176 }}</ref>  Linnik has shown that the number of ''p'' less than ''X'' such that ''n''(''p'') > X<sup>ε</sup> is bounded by a constant depending on ε.<ref name=FI156>{{cite book | title=Opera De Cribro | first1=John B. | last1=Friedlander | author1-link=John Friedlander | first2=Henryk  | last2=Iwaniec | author2-link=Henryk Iwaniec | publisher=[[American Mathematical Society]] | year=2010 | isbn=0-8218-4970-0 | zbl=1226.11099  | page=156 }}</ref>
+===Quadratic excess===
+Let ''p'' be an odd prime.  The '''quadratic excess''' ''E''(''p'') is the number of quadratic residues on the range (0,''p''/2) minus the number in the range (''p''/2,''p'').  For ''p'' congruent to 1 mod 4, the excess is zero, since −1 is a quadratic residue and the residues are symmetric under ''r'' ↔ ''p''−''r''.  For ''p'' congruent to 3 mod 4, the excess ''E'' is always positive.<ref>{{cite book | first1=Paul T. | last1=Bateman | author1-link=Paul T. Bateman | first2=Harold G. | last2=Diamond | title=Analytic Number Theory | publisher=World Scientific | year=2004 | isbn=981-256-080-7 | zbl=1074.11001 | page=250 }}</ref>
+==Complexity of finding square roots==
+That is, given a number ''a'' and a modulus ''n'', how hard is it
+# to tell whether an ''x'' solving ''x''<sup>2</sup> ≡ ''a'' (mod ''n'') exists
+# assuming one does exist, to calculate it?
+An important difference between prime and composite moduli shows up here. Modulo a prime ''p'', a quadratic residue ''a'' has 1 + (''a''|''p'')  roots (i.e. zero if ''a'' N ''p'', one if ''a'' ≡ 0 (mod ''p''), or two if ''a'' R ''p'' and gcd(''a,p'') = 1.)
+In general if a composite modulus ''n'' is written as a product of powers of distinct primes, and there are ''n''<sub>1</sub> roots modulo the first one, ''n''<sub>2</sub> mod the second, …, there will be ''n''<sub>1</sub>''n''<sub>2</sub>… roots modulo ''n''.
+The theoretical way solutions modulo the prime powers are combined to make solutions modulo ''n'' is called the [[Chinese remainder theorem]]; it can be implemented with an efficient algorithm.<ref>{{Harvnb|Bach|Shallit|1996|p=104 ff}}; it requires O(log<sup>2</sup> ''m'') steps where ''m'' is the number of primes dividing ''n''.</ref>
+<blockquote>For example:
+:Solve x<sup>2</sub> ≡ 6 (mod 15).
+::x<sup>2</sup> ≡ 6 (mod 3) has one solution, 0; x<sup>2</sup> ≡ 6 (mod 5) has two, 1 and 4.
+:: and there are two solutions modulo 15, namely 6 and 9.
+:Solve x<sup>2</sub> ≡ 4 (mod 15).
+::x<sup>2</sup> ≡ 4 (mod 3) has two solutions, 1 and 2; x<sup>2</sup> ≡ 4 (mod 5) has two, 2 and 3.
+:: and there are four solutions modulo 15, namely 2, 7, 8, and 13.
+</blockquote>
+===Prime or prime power modulus===
+First off, if the modulus ''n'' is prime the [[Legendre symbol]] (''a''|''n'') can be [[Jacobi symbol#Calculating the Jacobi symbol|quickly computed]] using a variation of [[Euclid's algorithm]].;<ref>{{Harvnb|Bach|Shallit|1996|p=113}}; computing (''a''|''n'') requires O(log ''a'' log ''n'') steps</ref> if it is &minus;1 there is no solution.
+Secondly, assuming that (''a''|''n'') = 1, if ''n'' ≡ 3 (mod 4), [[Joseph Louis Lagrange|Lagrange]] found that the solutions are given by
+:<math>x \equiv  \pm\; a^{(n+1)/4} \pmod{n},</math>
+and [[Adrien-Marie Legendre|Legendre]] found a similar solution<ref>Lemmermeyer, p. 29</ref> if ''n'' ≡ 5 (mod 8).
+For prime ''n'' ≡ 1 (mod 8), however, there is no known formula. [[Shanks&ndash;Tonelli algorithm|Tonelli]]<ref>{{Harvnb|Bach|Shallit|1996|p=156 ff}}; the algorithm requires O(log<sup>4</sup>''n'') steps.</ref> (in 1891) and [[Cipolla's algorithm|Cipolla]]<ref>{{Harvnb|Bach|Shallit|1996|p=156 ff}}; the algorithm requires O(log<sup>3</sup> ''n'') steps and is also nondetermisitic.</ref>  found  efficient algorithms that work for all prime moduli. Both algorithms require finding a quadratic nonresidue modulo ''n'', and there is no efficient deterministic algorithm known for doing that. But since half the numbers between 1 and ''n'' are nonresidues, picking numbers ''x'' at random and calculating the Legendre symbol (''x''|''n'') until a nonresidue is found will quickly produce one.
+If the modulus ''n'' is a prime power ''n'' = ''p''<sup>''e''</sup>, a solution may be found modulo ''p'' and "lifted" to a solution modulo ''n'' using [[Hensel's lemma]] or an algorithm of Gauss.<ref name="Gauss, DA, art. 101"/>
+===Composite modulus===
+If the modulus ''n'' has been factored into prime powers the solution was discussed above.
+If the Jacobi symbol (''a''|''n'') = &minus;1 then there is no solution. If it is +1, there may or may not be one.
+If the factorization of ''n'' is not known, and (''a''|''n'') = 1, the problem is known to be equivalent to [[integer factorization]] of ''n'' (i.e. an efficient solution to either problem could be used to solve the other efficiently).
+<blockquote>The above discussion indicates how knowing the factors of ''n'' allows us to find the roots efficiently. Say there were an efficient algorithm for finding square roots modulo a composite number. The article [[congruence of squares]] discusses how finding two numbers x and y where ''x''<sup>2</sup> ≡ ''y''<sup>2</sup> (mod ''n'') and ''x'' &ne; ±''y'' suffices to factorize ''n'' efficiently. Generate a random number, square it modulo ''n'', and have the efficient square root algorithm find a root. Repeat until it returns a number not equal to the one we originally squared (or its negative modulo ''n''), then follow the algorithm described in congruence of squares. The efficiency  of the factoring algorithm depends on the exact characteristics of the root-finder (e.g. does it return all roots? just the smallest one? a random one?), but it will be efficient.<ref>Crandall & Pomerance, ex. 6.5 & 6.6, p.273</ref></blockquote>
+Simply determining whether ''a'' N ''n'' or ''a'' R ''n'' (which can be done efficiently for prime ''n'' by computing the Legendre symbol) is known as the [[quadratic residuosity problem]] when ''n'' is composite. It is not known to be as [[Computational hardness assumption|hard]] as factorization, but is thought to be quite hard.
+On the other hand, if we want to know if there is a solution for ''x'' less than some given limit ''c'', this problem is [[NP-complete]];<ref>{{harvnb|Manders|Adleman|1978}}</ref> however, this is a [[fixed-parameter tractable]] problem, where ''c'' is the parameter.
+In general, to determine if a quadratic congruence with composite modulus is solvable use the following theorem:<ref>{{cite book|last=Burton|first=David|title=Elementary Number Theory|year=2007|publisher=McGraw HIll|location=New York|pages=195}}</ref>
+Let ''n'' > 1, and gcd(''a'',''n'') =1. Then ''x''<sup>2</sup> ≡ ''a'' (mod ''n'') is solvable if and only if:
+a) The Legendre symbol, (''a''/''p'') = 1 for all odd prime divisors of ''n''.
+b) ''a'' ≡ 1 (mod 4) if 4|n, but 8 does not divide ''n''; ''a'' ≡ 1 (mod 8) if 8|''n''.
+Note: This theorem essentially requires that the factorization of ''n'' is known.  Also notice that if gcd(''a'',''n'')=''m'', then the congruence can be reduced to ''a''/''m'' ≡ ''x''<sup>2</sup>/''m'' (mod ''n''/''m''), but then this takes the problem away from being a problem of quadratic congruences (unless ''m'' is a square).
+==The number of quadratic residues==
+The list of the number of quadratic residues mod ''n'', for ''n''=1,2,3 ..., looks like:<ref>{{OEIS link|A000224}}</ref>
+: 1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, 4, 9, 8, 10, 6, 8, 12, 12, 6, 11, 14, 11, 8, 15, 12, 16, 7, 12, 18, 12, 8, 19, 20, 14, 9, 21, 16, 22, 12, 12, 24, 24, 8, 22, 22, 18, 14, 27, 22, 18, 12, 20, 30, 30, 12, 31, 32, 16, 12, 21, ...
+A formula to count the number of squares mod ''n'' is given by Stangl.<ref>{{Citation |last=Stangl |first=Walter D. |title=Counting Squares in &#8484;<sub>''n''</sub> |journal=Mathematics Magazine |volume=69 |issue=4 |pages=285&ndash;289 |date=October 1996 |url=http://mathdl.maa.org/images/cms_upload/Walter_D22068._Stangl.pdf |doi= }}</ref>
+==Applications of quadratic residues==
+===Acoustics===
+[[Diffusion (acoustics)#Quadratic-Residue Diffusors|Sound diffusers]] have been based on number-theoretic concepts such as [[primitive root modulo n|primitive roots]] and quadratic residues.<ref>{{cite web|last=Walker|first=R|title=The design and application of modular acoustic diffusing elements|url=http://www.bbc.co.uk/rd/pubs/reports/1990-15.pdf|publisher=BBC Research Department|accessdate=3 June 2012}}</ref>
+===Graph theory===
+[[Paley graph]]s are dense undirected graphs, one for each prime ''p'' ≡ 1 (mod 4), that  form an infinite family of [[conference graph]]s, which yield an infinite family of [[symmetric matrix|symmetric]] [[conference matrix|conference matrices]].
+Paley digraphs are directed analogs of Paley graphs, one for each ''p'' ≡ 3 (mod 4), that yield [[Skew-symmetric matrix|antisymmetric]] conference matrices.
+The construction of these graphs uses quadratic residues.
+===Cryptography===
+The fact that finding a square root of a number modulo a large composite ''n'' is equivalent to factoring (which is widely believed be a hard problem) has been used for constructing [[cryptography|cryptographic schemes]] such as the [[Rabin cryptosystem]] and the [[oblivious transfer]]. The [[quadratic residuosity problem]] is the basis for the [[Goldwasser-Micali cryptosystem]].
+The [[discrete logarithm]] is a similar problem that is also used in cryptography.
+===Primality testing===
+[[Euler's criterion]] is a formula for the Legendre symbol (''a''|''p'') where ''p'' is prime. If ''p'' is composite the formula may or may not compute (''a''|''p'') correctly. The [[Solovay-Strassen primality test]] for whether a given number ''n'' is prime or composite picks a random ''a'' and computes (''a''|''n'') using a modification of Euclid's algorithm,<ref>{{Harvnb|Bach|Shallit|1996|p=113}}</ref> and also using Euler's criterion.<ref>{{Harvnb|Bach|Shallit|1996|pp=109&ndash;110}}; Euler's criterion requires O(log<sup>3</sup> ''n'') steps</ref> If the results disagree, ''n'' is composite; if they agree, ''n'' may be composite or prime. For a composite ''n'' at least 1/2 the values of ''a'' in the range 2, 3, ...,  ''n'' &minus; 1 will return "''n'' is composite"; for prime ''n'' none will. If, after using many different values of ''a'', ''n'' has not been proved composite it is called a "probable prime".
+The [[Miller-Rabin primality test]] is based on the same principles. There is a deterministic version of it, but the proof that it works depends on the [[generalized Riemann hypothesis]]; the output from this test is "''n'' is definitely composite" or "either ''n'' is prime or the GRH is false". If the second output ever occurs for a composite ''n'', then the GRH would be false, which would have implications through many branches of mathematics.
+===Integer factorization===
+In § VI of the ''Disquisitiones Arithmeticae''<ref>Gauss, DA, arts 329&ndash;334</ref> Gauss discusses two factoring algorithms that use quadratic residues and the [[law of quadratic reciprocity]].
+Several modern factorization algorithms (including [[Dixon's algorithm]], the [[continued fraction factorization|continued fraction method]], the [[quadratic sieve]], and the [[General number field sieve|number field sieve]]) generate small quadratic residues (modulo the number being factorized) in an attempt to find a [[congruence of squares]] which will yield a factorization. The number field sieve is the fastest general-purpose factorization algorithm known.
+==Table of quadratic residues==
+{| class="wikitable" style="text-align:center;"
+|+ Quadratic Residues
+|-
+! x || 1||2|| 3|| 4||5|| 6|| 7|| 8 ||9 ||10|| 11|| 12|| 13|| 14 ||15|| 16|| 17|| 18 ||19|| 20|| 21|| 22|| 23|| 24 ||25
+|-
+! x<sup>2</sup>
+! style="width:25px;"|1
+! style="width:25px;"|4
+! style="width:25px;"|9
+! style="width:25px;"|16
+! style="width:25px;"|25
+! style="width:25px;"|36
+! style="width:25px;"|49
+! style="width:25px;"|64
+! style="width:25px;"|81
+!100|| 121|| 144|| 169|| 196 ||225|| 256|| 289|| 324 ||361 || 400|| 441|| 484|| 529|| 576 ||625
+|-
+! mod 2
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1
+|-
+! mod 3
+| 1 || 1 || 0
+| 1 || 1 || 0
+| 1 || 1 || 0
+| 1 || 1 || 0
+| 1 || 1 || 0
+| 1 || 1 || 0
+| 1 || 1 || 0
+| 1 || 1 || 0
+| 1
+|-
+! mod 4
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1 || 0
+| 1
+|-
+! mod 5
+| 1 || 4 || 4 || 1 || 0
+| 1 || 4 || 4 || 1 || 0
+| 1 || 4 || 4 || 1 || 0
+| 1 || 4 || 4 || 1 || 0
+| 1 || 4 || 4 || 1 || 0
+|-
+! mod 6
+| 1 || 4 || 3 || 4 || 1 || 0
+| 1 || 4 || 3 || 4 || 1 || 0
+| 1 || 4 || 3 || 4 || 1 || 0
+| 1 || 4 || 3 || 4 || 1 || 0
+| 1
+|-
+! mod 7
+| 1 || 4 || 2 || 2 || 4 || 1 || 0
+| 1 || 4 || 2 || 2 || 4 || 1 || 0
+| 1 || 4 || 2 || 2 || 4 || 1 || 0
+| 1 || 4 || 2 || 2
+|-
+! mod 8
+| 1 || 4 || 1 || 0
+| 1 || 4 || 1 || 0
+| 1 || 4 || 1 || 0
+| 1 || 4 || 1 || 0
+| 1 || 4 || 1 || 0
+| 1 || 4 || 1 || 0
+| 1
+|-
+! mod 9
+| 1 || 4 || 0 || 7 || 7 || 0 || 4 || 1 || 0
+| 1 || 4 || 0 || 7 || 7 || 0 || 4 || 1 || 0
+| 1 || 4 || 0 || 7 || 7 || 0 || 4
+|-
+! mod 10
+| 1 || 4 || 9 || 6 || 5 || 6 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 6 || 5 || 6 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 6 || 5
+|-
+! mod 11
+| 1 || 4 || 9 || 5 || 3 || 3 || 5 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 5 || 3 || 3 || 5 || 9 || 4 || 1 || 0
+| 1 || 4 || 9
+|-
+! mod 12
+| 1 || 4 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 4 || 1 || 0
+| 1
+|-
+! mod 13
+| 1 || 4 || 9 || 3 || 12 || 10 || 10 || 12 || 3 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 3 || 12 || 10 || 10 || 12 || 3 || 9 || 4 || 1
+|-
+! mod 14
+| 1 || 4 || 9 || 2 || 11 || 8 || 7 || 8 || 11 || 2 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 2 || 11 || 8 || 7 || 8 || 11 || 2 || 9
+|-
+! mod 15
+| 1 || 4 || 9 || 1 || 10 || 6 || 4 || 4 || 6 || 10 || 1 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 1 || 10 || 6 || 4 || 4 || 6 || 10
+|-
+! mod 16
+| 1 || 4 || 9 || 0 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 0 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 0 || 9 || 4 || 1 || 0
+| 1
+|-
+! mod 17
+| 1 || 4 || 9 || 16 || 8 || 2 || 15 || 13 || 13 || 15 || 2 || 8 || 16 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 16 || 8 || 2 || 15 || 13
+|-
+! mod 18
+| 1 || 4 || 9 || 16 || 7 || 0 || 13 || 10 || 9 || 10 || 13 || 0 || 7 || 16 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 16 || 7 || 0 || 13
+|-
+! mod 19
+| 1 || 4 || 9 || 16 || 6 || 17 || 11 || 7 || 5 || 5 || 7 || 11 || 17 || 6 || 16 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 16 || 6 || 17
+|-
+! mod 20
+| 1 || 4 || 9 || 16 || 5 || 16 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 16 || 5 || 16 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 16 || 5
+|-
+! mod 21
+| 1 || 4 || 9 || 16 || 4 || 15 || 7 || 1 || 18 || 16 || 16 || 18 || 1 || 7 || 15 || 4 || 16 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 16
+|-
+! mod 22
+| 1 || 4 || 9 || 16 || 3 || 14 || 5 || 20 || 15 || 12 || 11 || 12 || 15 ||  20 || 5 || 14 || 3 || 16 ||  9 || 4 || 1 || 0
+| 1 || 4 || 9
+|-
+! mod 23
+| 1 || 4 || 9 || 16 || 2 || 13 || 3 || 18 || 12 || 8 || 6 || 6 || 8 || 12 ||18 || 3 || 13 || 2 || 16 || 9 || 4 || 1 || 0
+| 1 || 4
+|-
+! mod 24
+| 1 || 4 || 9 || 16 || 1 || 12 || 1 || 16 || 9 || 4 || 1 || 0
+| 1 || 4 || 9 || 16 || 1 || 12 || 1 || 16 || 9 || 4 || 1 || 0
+| 1
+|-
+! mod 25
+| 1 || 4 || 9 || 16 || 0 || 11 || 24 || 14 || 6 || 0 || 21 || 19  || 19 || 21 || 0 || 6 || 14 || 24 || 11 || 0 || 16 || 9 || 4 || 1 || 0
+|}
+==See also==
+*[[Euler's criterion]]
+*[[Gauss's lemma (number theory)|Gauss's lemma]]
+*[[Zolotarev's lemma]]
+*[[Character sum]]
+*[[Law of quadratic reciprocity]]
+*[[Quadratic residue code]]
+==Notes==
+{{reflist|colwidth=30em}}
+==References==
+The ''[[Disquisitiones Arithmeticae]]'' has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
+*{{citation
+  | last1 = Gauss  | first1 = Carl Friedrich
+  | last2 = Clarke | first2 = Arthur A. (translator into English)
+  | title = Disquisitiones Arithemeticae |edition=Second corrected
+  | publisher = [[Springer Science+Business Media|Springer]]
+  | location = New York
+  | year = 1986
+  | isbn = 0-387-96254-9}}
+*{{citation
+  | last1 = Gauss | first1 = Carl Friedrich
+  | last2 = Maser | first2 = H. (translator into German)
+  | title = Untersuchungen über hohere Arithmetik |trans_title=Disquisitiones Arithemeticae & other papers on number theory |edition=second
+  | publisher = Chelsea
+  | location = New York
+  | year = 1965
+  | isbn = 0-8284-0191-8}}
+*{{citation
+  | last1 = Bach  | first1 = Eric
+  | last2 = Shallit | first2 = Jeffrey
+  | series = Algorithmic Number Theory |volume=I |title=Efficient Algorithms
+  | publisher = [[The MIT Press]]
+  | location = Cambridge
+  | year = 1996
+  | isbn = 0-262-02405-5}}
+*{{citation
+  | last1 = Crandall  | first1 = Richard
+  | last2 = Pomerance  | first2 = Carl
+  | title = Prime Numbers: A Computational Perspective
+  | publisher = Springer
+  | location = New York
+  | year = 2001
+  | isbn = 0-387-94777-9}}
+*{{Citation
+  | last = Davenport
+  | first = Harold
+  | title = Multiplicative Number Theory |edition=third
+  | publisher = Springer
+  | location = New York
+  | year = 2000
+  | isbn = 0-387-95097-4}}
+*{{Citation
+  |last=Garey |first= Michael R. |authorlink=Michael R. Garey |last2=Johnson |first2=David S. |author2-link=David S. Johnson
+  | year = 1979
+  | title = Computers and Intractability: A Guide to the Theory of NP-Completeness
+  | publisher = W. H. Freeman
+  | isbn = 0-7167-1045-5}} A7.1: AN1, pg.249.
+*{{citation
+  | last1 = Hardy  | first1 = G. H.
+  | last2 = Wright | first2 = E. M.
+  | title = An Introduction to the Theory of Numbers |edition=fifth
+  | publisher = [[Oxford University Press]]
+  | location = Oxford
+  | year = 1980
+  | isbn = 978-0-19-853171-5}}
+*{{citation
+  | last1 = Ireland  | first1 = Kenneth
+  | last2 = Rosen  | first2 = Michael
+  | title = A Classical Introduction to Modern Number Theory |edition=second
+  | publisher = Springer
+  | location = New York
+  | year = 1990
+  | isbn = 0-387-97329-X}}
+*{{citation
+  | last1 = Lemmermeyer  | first1 = Franz
+  | title = Reciprocity Laws: from Euler to Eisenstein
+  | publisher = Springer
+  | location = Berlin
+  | year = 2000
+  | isbn = 3-540-66957-4}}
+*{{Citation
+  |first=Kenneth L. |last=Manders
+  |last2=Adleman |first2=Leonard |author2-link=Leonard Adleman
+  |title = ''NP''-Complete Decision Problems for Binary Quadratics
+  |journal = Journal of Computer and System Sciences
+  |volume=16 |issue=2 |year=1978
+  |pages = 168&ndash;184
+  |doi = 10.1016/0022-0000(78)90044-2
+  |postscript = .}}
+==External links==
+*{{MathWorld|urlname=QuadraticResidue|title=Quadratic Residue}}
+*{{PlanetMath|urlname=PolyaVinogradovInequality|title=Proof of Pólya&ndash;Vinogradov inequality}}
+{{DEFAULTSORT:Quadratic Residue}}
+[[Category:Modular arithmetic]]
+[[Category:NP-complete problems]]
+[[Category:Quadratic residue]]
+[[ja:平方剰余]]
+[[zh-classical:二次剩餘]]

Angular frequency: Difference between revisions

Revision as of 18:59, 11 November 2013

History, conventions, and elementary facts

Prime modulus

Prime power modulus

Composite modulus not a prime power

Notations

Distribution of quadratic residues

Dirichlet's formulas

Pairs of residues and nonresidues

The Pólya–Vinogradov inequality

Least quadratic non-residue

Quadratic excess

Complexity of finding square roots

Prime or prime power modulus

Composite modulus

The number of quadratic residues

Applications of quadratic residues

Acoustics

Graph theory

Cryptography

Primality testing

Integer factorization

Table of quadratic residues

See also

Notes

References

External links

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