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| | Let's look an actual registry scan plus a few of what you will see whenever we do 1 on a computer. This test was done on a computer which was not working as it could, running at slow speed and having certain issues with freezing up.<br><br>If you registry gets cluttered up with a great deal of junk we don't use, your PC can run slower. Therefore it's prudent that you regularly get a registry cleaned.<br><br>Although this issue affects millions of computer users throughout the globe, there is an easy method to fix it. You see, there's one reason for a slow loading computer, plus that's considering a PC cannot read the files it needs to run. In a nutshell, this simply means which whenever you do anything on Windows, it requirements to read up on how to do it. It's traditionally a pretty 'dumb' system, that has to have files to tell it to do everything.<br><br>If you feel we don't have enough money at the time to upgrade, then the number one choice is to free up several room by deleting a few of the unwanted files and folders.<br><br>Another thing we should check is whether or not the [http://bestregistrycleanerfix.com/regzooka regzooka] program which you are considering has the ability to identify files and programs which are wise. One of the registry cleaner programs we might try is RegCure. It is helpful for speeding up and cleaning up issues on a computer.<br><br>2)Fix a Windows registry to speed up PC- The registry is a complex section of your computer which holds different kinds of data from the factors you do on your computer daily. Coincidentally, over time the registry might become cluttered with info and/or can receive certain kind of virus. This is especially important and you MUST get this problem fixed right away, otherwise we run the risk of the computer being permanently damage and/or the sensitive information (passwords, etc.) is stolen.<br><br>The first reason the computer will be slow is because it needs more RAM. You'll see this issue right away, especially should you have lower than a gig of RAM. Most brand-new computers come with a least that much. While Microsoft claims Windows XP will run on 128 MB, it and Vista really need at least a gig to run smoothly and enable we to run numerous programs at once. Fortunately, the cost of RAM has dropped greatly, and there are a gig installed for $100 or less.<br><br>Next, there is an easy means to deal with this issue. You are able to install a registry cleaner which there are it found on the internet. This software can help you see out these errors inside a computer and clean them. It equally could figure out these malware and other threats that influence the speed of the computer. So this software will speed up PC easier. You are able to choose one of these methods to accelerate we computer. |
| In [[mathematics]], a '''Diophantine equation''' is a [[polynomial]] [[equation]] in two or more [[unknown]]s such that only the [[integer]] solutions are searched or studied (an integer solution is a solution such that all the unknowns take integer values). A '''linear Diophantine equation''' is an equation between two sums of [[monomials]] of [[Degree of a polynomial|degree]] zero or one.
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| '''Diophantine problems''' have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an [[algebraic curve]], [[algebraic surface]], or more general object, and ask about the [[lattice point]]s on it.
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| The word ''Diophantine'' refers to the [[Hellenistic civilization|Hellenistic]] mathematician of the 3rd century, [[Diophantus]] of [[Alexandria]], who made a study of such equations and was one of the first mathematicians to introduce [[mathematical symbol|symbolism]] into [[algebra]]. The mathematical study of Diophantine problems that Diophantus initiated is now called '''Diophantine analysis'''.
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| While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (beyond the theory of [[quadratic form]]s) was an achievement of the twentieth century.
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| ==Examples of Diophantine equations==
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| {|
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| |-
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| | align="left" colspan=2 | In the following Diophantine equations, ''x'', ''y'', and ''z'' are the unknowns and the other letters are given constants.
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| | <math>ax+by=1\,</math>||This is a linear Diophantine equation (see the [[#Linear Diophantine equations|section "Linear Diophantine equations"]] below).
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| | <math>x^n+y^n=z^n \,</math>||For ''n'' = 2 there are infinitely many solutions (''x'',''y'',''z''): the [[Pythagorean triple]]s. For larger integer values of ''n'', [[Fermat's Last Theorem]] states there are no positive integer solutions (''x'', ''y'', ''z'').
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| | <math>x^2-ny^2=\pm 1\,</math>|| This is [[Pell's equation]], which is named after the English mathematician [[John Pell]]. It was studied by [[Brahmagupta]] in the 7th century, as well as by [[Pierre de Fermat|Fermat]] in the 17th century.
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| | <math>\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}</math>||The [[Erdős–Straus conjecture]] states that, for every positive integer ''n'' ≥ 2, there exists a solution in ''x'', ''y'', and ''z'', all as positive integers. Although not usually stated in polynomial form, this example is equivalent to the polynomial equation 4''xyz'' = ''yzn'' + ''xzn'' + ''xyn'' = ''n''(''yz'' + ''xz'' + ''xy'').
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| |}
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| =={{anchor|Linear Diophantine}}Linear Diophantine equations==
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| The simplest linear Diophantine equation takes the form ''ax'' + ''by'' = ''c'', where ''a'', ''b'' and ''c'' are given integers. The solutions are completely described by the following theorem: ''This Diophantine equation has a solution'' (where ''x'' and ''y'' are integers) ''if and only if'' ''c'' ''is a multiple of the [[greatest common divisor]] of'' ''a'' ''and'' ''b''. ''Moreover, if'' (''x'', ''y'') ''is a solution, then the other solutions have the form'' (''x'' + ''kv'', ''y'' - ''ku''), ''where'' ''k'' ''is an arbitrary integer, and'' ''u'' ''and'' ''v'' ''are the quotients of'' ''a'' ''and'' ''b'' ''(respectively) by the greatest common divisor of'' ''a'' ''and'' ''b''.
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| ''Proof:'' If ''d'' is this greatest common divisor, [[Bézout's identity]] asserts the existence of integers ''e'' and ''f'' such that ''ae'' + ''bf'' = ''d''. If ''c'' is a multiple of ''d'', then ''c'' = ''dh'' for some integer ''h'', and (''eh'', ''fh'') is a solution. On the other hand, for every integers ''x'' and ''y'', the greatest common divisor ''d'' of ''a'' and ''b'' divides ''ax'' + ''by''. Thus, if the equation has a solution, then ''c'' must be a multiple of ''d''. If ''a'' = ''ud'' and ''b'' = ''vd'', then for every solution (''x'', ''y''), we have
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| :<math> a(x+kv)+ b(y-ku) = ax+by + k(av -bu) =ax+by + k(udv -vdu) =ax + by</math>,
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| showing that (''x'' + ''kv'', ''y'' - ''ku'') is another solution. Finally, given two solutions such that {{nowrap|1=''ax<sub>1</sub>'' + ''by<sub>1</sub>'' = ''ax<sub>2</sub>'' + ''by<sub>2</sub>'' = ''c''}}, one deduces that {{nowrap|1=''u'' (''x''<sub>2</sub> - ''x''<sub>1</sub>) + ''v'' (''y''<sub>2</sub> - ''y''<sub>1</sub>) = 0}}. As ''u'' and ''v'' are [[coprime]], [[Euclid's lemma]] shows that there exists an integer ''k'' such that {{nowrap|1=''x''<sub>2</sub> - ''x''<sub>1</sub> = ''kv''}} and {{nowrap|1=''y''<sub>2</sub> - ''y''<sub>1</sub> = -''ku''}}. Therefore {{nowrap|1=''x''<sub>2</sub> = ''x''<sub>1</sub> + ''kv''}} and {{nowrap|1=''y''<sub>2</sub> = ''y''<sub>1</sub> - ''kv''}}, which completes the proof.
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| The [[Chinese remainder theorem]] describes an important class of linear Diophantine systems of equations: let ''n''<sub>1</sub>, ..., ''n''<sub>''k''</sub> be ''k'' [[pairwise coprime]] integers greater than one, ''a''<sub>1</sub>, ..., ''a''<sub>''k''</sub> be ''k'' arbitrary integers, and ''N'' be the product ''n''<sub>1</sub> ··· ''n''<sub>''k''</sub>. The Chinese remainder theorem asserts that the following linear Diophantine system has exactly one solution {{nowrap|(''x'', ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>)}} such that {{nowrap|1=0 ≤ ''x'' < ''N''}}, and that the other solutions are obtained by adding to ''x'' a multiple of ''N'':
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| :<math>\begin{align}
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| x&= a_1 + n_1\,x_1\\
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| &\cdots\\
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| x&=a_k+n_k\,x_k
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| \end{align}</math>
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| More generally, every system of linear Diophantine equation may be solved by computing the [[Smith normal form]] of its matrix, in a way that is similar to the use of the [[Reduced row echelon form]] to solve a [[system of linear equations]] over a field. Using using [[matrix (mathematics)|matrix notation]] every system of linear Diophantine equations may be written
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| :<math>A\,X=C,</math>
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| where {{math|''A''}} is a ''m''×''n'' matrix of integers, {{math|''X''}} is a ''n''×1 [[column matrix]] of unknowns and {{math|''C''}} is a ''m''×1 column matrix of integers.
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| The computation of the Smith normal form of {{math|''A''}} provides two [[unimodular matrix|unimodular matrices]] (that is matrices that are invertible over the integers, which have ±1 as determinant) {{math|''U''}} and {{math|''V''}} of respective dimensions ''m''×''m'' and ''n''×''n'', such that the matrix
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| :<math>B=\left[b_{i,j}\right]=UAV</math>
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| is such that {{nowrap|''b''<sub>''i'',''i''</sub>}} is not zero for ''i'' not greater than some integer ''k'', and all the other entries are zero. The system to be solved may thus be rewritten as | |
| :<math>B\,(V^{-1}X) = UC.</math>
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| Calling {{math|''y''<sub>''i''</sub>}} the entries of <math>V^{-1}X</math> and {{math|''d''<sub>''i''</sub>}} those of <math>UC,</math> this leads to the system
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| :{{math|1=''b''<sub>''i'',''i''</sub> ''y''<sub>''i''</sub> = ''d''<sub>''i''</sub>}} for 1 ≤ ''i'' ≤ ''k'',
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| :{{math|1=0 ''y''<sub>''i''</sub> = ''d''<sub>''i''</sub>}} for ''k'' < ''i'' ≤ ''n''.
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| This system is equivalent to the given one in the following sense: A column matrix of integers {{math|''x''}} is a solution of the given system if and only {{math|1= ''x'' = ''V y''}} for some column matrix of integers {{math|''y''}} such that {{math|1=''By'' = ''D''}}.
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| It follows that the system has a solution if and only if {{math|1=''b''<sub>''i'',''i''</sub>}} divides {{math|1=''d''<sub>''i'',''i''</sub>}} for ''i'' ≤ ''k'' and {{math|1=''d''<sub>''i''</sub> = 0}} for ''i'' > ''k''. If this condition is fulfilled, the solutions of the given system are
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| :<math> U\,\left[
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| \begin{array}{l}
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| \frac{d_1}{b_{1,1}}\\
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| \cdots\\
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| \frac{d_k}{b_{k,k}}\\
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| h_{k+1}\\
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| \cdots\\
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| h_n
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| \end{array}
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| \right]\,, </math>
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| where {{math|''h''<sub>''k''+1</sub>, ..., ''h''<sub>''n''</sub>}} are arbitrary integers.
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| ==Diophantine analysis==
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| === Typical questions ===
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| The questions asked in Diophantine analysis include:
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| #Are there any solutions?
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| #Are there any solutions beyond some that are easily found by inspection?
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| #Are there finitely or infinitely many solutions?
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| #Can all solutions be found in theory?
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| #Can one in practice compute a full list of solutions?
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| These traditional problems often lay unsolved for centuries, and mathematicians gradually came to understand their depth (in some cases), rather than treat them as puzzles.
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| === Typical problem ===
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| <nowiki> The given information that a father's age is 1 less than twice that of his son, and that the digits AB making up the father's age are reversed in the son's age (i.e. BA), leads to the equation 19B - 8A = 1. Inspection gives the result 73 and 37 years.</nowiki>
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| ===17th and 18th centuries===
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| In 1637, [[Pierre de Fermat]] scribbled on the margin of his copy of ''[[Arithmetica]]'': "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers." Stated in more modern language, "The equation ''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup> has no solutions for any ''n'' higher than 2." And then he wrote, intriguingly: "I have discovered a truly marvelous proof of this proposition, which this margin is too narrow to contain." Such a proof eluded mathematicians for centuries, however, and as such his statement became famous as [[Fermat's Last Theorem]]. It wasn't until 1994 that it was proven by the British mathematician [[Andrew Wiles]].
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| In 1657, Fermat attempted to solve the Diophantine equation 61''x''<sup>2</sup> + 1 = ''y''<sup>2</sup> (solved by [[Brahmagupta]] over 1000 years earlier). The equation was eventually solved by [[Euler]] in the early 18th century, who also solved a number of other Diophantine equations.
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| ===Hilbert's tenth problem===
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| In 1900, in recognition of their depth, [[David Hilbert]] proposed the solvability of all Diophantine problems as [[Hilbert's tenth problem|the tenth]] of his [[Hilbert's problems|celebrated problems]]. In 1970, a novel result in [[mathematical logic]] known as [[Matiyasevich's theorem]] settled the problem negatively: in general Diophantine problems are unsolvable.
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| [[Diophantine geometry]], which is the application of techniques from [[algebraic geometry]] in this field, has continued to grow as a result; since treating arbitrary equations is a dead end, attention turns to equations that also have a geometric meaning. The central idea of Diophantine geometry is that of a [[rational point]], namely a solution to a polynomial equation or a [[system of polynomial equations]], which is a vector in a prescribed [[field (mathematics)|field]] ''K'', when ''K'' is ''not'' [[algebraically closed]].
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| ===Modern research===
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| One of the few general approaches is through the [[Hasse principle]]. [[Infinite descent]] is the traditional method, and has been pushed a long way.
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| The depth of the study of general Diophantine equations is shown by the characterisation of [[Diophantine set]]s as equivalently described as [[recursively enumerable set|recursively enumerable]]. In other words, the general problem of Diophantine analysis is blessed or cursed with universality, and in any case is not something that will be solved except by re-expressing it in other terms.
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| The field of [[Diophantine approximation]] deals with the cases of ''Diophantine inequalities''. Here variables are still supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper and lower bounds.
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| The most celebrated single question in the field, the [[conjecture]] known as [[Fermat's Last Theorem]], was solved by [[Andrew Wiles]]<ref>[http://www.pbs.org/wgbh/nova/proof/wiles.html Solving Fermat: Andrew Wiles]</ref> but using tools from algebraic geometry developed during the last century rather than within number theory where the conjecture was originally formulated. Other major results, such as [[Faltings' theorem]], have disposed of old conjectures.
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| ===Infinite Diophantine equations===
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| An example of an infinite diophantine equation is:
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| :<math>
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| N = A^2+2B^2+3C^2+4D^2+5E^2+....
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| </math>
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| which can be expressed as "How many ways can a given integer ''N'' be written as the sum of a square plus twice a square plus thrice a square and so on?" The number of ways this can be done for each ''N'' forms an integer sequence. Infinite Diophantine equations are related to [[theta functions]] and infinite dimensional lattices. This equation always has a solution for any positive ''N''. Compare this to:
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| :<math>
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| N = A^2+4B^2+9C^2+16D^2+25E^2+....
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| </math>
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| which does not always have a solution for positive ''N''. | |
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| ==Exponential Diophantine equations==
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| If a Diophantine equation has as an additional variable or variables occurring as [[exponentiation|exponents]], it is an exponential Diophantine equation. One example is the [[Ramanujan–Nagell equation]], 2<sup>''n''</sup> − 7 = ''x''<sup>2</sup>. Such equations do not have a general theory; particular cases such as [[Catalan's conjecture]] have been tackled. However, the majority are solved via ad hoc methods such as [[Størmer's theorem]] or even [[trial and error]].
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{Cite book| first=L. J.|last=Mordell | authorlink=Louis Mordell | title=Diophantine equations | publisher=[[Academic Press]] | year=1969 | isbn=0-12-506250-8 }}
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| * {{Cite book|last=Schmidt|first=Wolfgang M.|authorlink=Wolfgang M. Schmidt|title=Diophantine approximations and Diophantine equations|series=Lecture Notes in Mathematics|publisher=[[Springer-Verlag]]|year=2000}}
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| * {{Cite book| first1=T. N.|last1=Shorey | first2=R.|last2=Tijdeman |author2-link=Robert Tijdeman| title=Exponential Diophantine equations | series=Cambridge Tracts in Mathematics | volume=87 | publisher=[[Cambridge University Press]] | year=1986 | isbn=0-521-26826-5 }}
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| * {{Cite book| first=N. P.|last=Smart | title=The algorithmic resolution of Diophantine equations | series=London Mathematical Society Student Texts | volume=41 | publisher=Cambridge University Press | year=1998 | isbn=0-521-64156-X }}
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| *{{Cite book | first=John | last=Stillwell | title=Mathematics and its History | edition=Second Edition | publisher=Springer Science + Business Media Inc. | year=2004 | isbn=0-387-95336-1 }}
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| ==External links==
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| *[http://mathworld.wolfram.com/DiophantineEquation.html Diophantine Equation]. From [[MathWorld]] at [[Wolfram Research]].
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| *[http://planetmath.org/encyclopedia/DiophantineEquation.html Diophantine Equation]. From [[PlanetMath]].
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| *{{Springer|id=d/d032610|title=Diophantine equations}}
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| *[http://www.alpertron.com.ar/QUAD.HTM Dario Alpern's Online Calculator]. Retrieved 18 March 2009
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| {{DEFAULTSORT:Diophantine Equation}}
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| [[Category:Diophantine equations|*]]
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Let's look an actual registry scan plus a few of what you will see whenever we do 1 on a computer. This test was done on a computer which was not working as it could, running at slow speed and having certain issues with freezing up.
If you registry gets cluttered up with a great deal of junk we don't use, your PC can run slower. Therefore it's prudent that you regularly get a registry cleaned.
Although this issue affects millions of computer users throughout the globe, there is an easy method to fix it. You see, there's one reason for a slow loading computer, plus that's considering a PC cannot read the files it needs to run. In a nutshell, this simply means which whenever you do anything on Windows, it requirements to read up on how to do it. It's traditionally a pretty 'dumb' system, that has to have files to tell it to do everything.
If you feel we don't have enough money at the time to upgrade, then the number one choice is to free up several room by deleting a few of the unwanted files and folders.
Another thing we should check is whether or not the regzooka program which you are considering has the ability to identify files and programs which are wise. One of the registry cleaner programs we might try is RegCure. It is helpful for speeding up and cleaning up issues on a computer.
2)Fix a Windows registry to speed up PC- The registry is a complex section of your computer which holds different kinds of data from the factors you do on your computer daily. Coincidentally, over time the registry might become cluttered with info and/or can receive certain kind of virus. This is especially important and you MUST get this problem fixed right away, otherwise we run the risk of the computer being permanently damage and/or the sensitive information (passwords, etc.) is stolen.
The first reason the computer will be slow is because it needs more RAM. You'll see this issue right away, especially should you have lower than a gig of RAM. Most brand-new computers come with a least that much. While Microsoft claims Windows XP will run on 128 MB, it and Vista really need at least a gig to run smoothly and enable we to run numerous programs at once. Fortunately, the cost of RAM has dropped greatly, and there are a gig installed for $100 or less.
Next, there is an easy means to deal with this issue. You are able to install a registry cleaner which there are it found on the internet. This software can help you see out these errors inside a computer and clean them. It equally could figure out these malware and other threats that influence the speed of the computer. So this software will speed up PC easier. You are able to choose one of these methods to accelerate we computer.