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| {{expert-subject|mathematics|date=January 2011}}
| | Make sure you can hear your breathe (except in public places where you may want to be more discreet) and you can see the stomach moving as this will help you to build up a rhythm. A little more than 30 miles, with a 5,200 foot elevation gain, and you'll ride over the finish line. In case you adored this informative article and also you want to obtain more details with regards to [http://wallpaper.lazonegeek.com/profile/sapeltier Popular mountain bike sizing.] i implore you to stop by the page. The contemporary bikes for mountains are provided with stronger and lighter frame types in addition to pioneering form and design. There are more than 15 circuits in the sport of professional mountain biking. If you've ridden for years and years and know what you need, buying online does make sense. <br><br> |
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| In mathematics, the '''Pocklington–Lehmer primality test''' is a [[primality test]] devised by [[Henry Cabourn Pocklington]] and [[Derrick Henry Lehmer]] to decide whether a given number <math>N</math> is prime. The output of the test is a proof that the number is prime or that primality could not be established.
| | As hardtails have front-only suspension, they have less pedal bob and increased pedal stroke efficiency when compared to dual suspension MTB bikes. You can install an electric motor on your regular bike in about an hour or hire a bike mechanic to do it. An alternative to Clipless pedals are Mountain Bike shoes with Shimano Pedaling Dynamics (SPD). He or she doesn't have to get into an accident just to understand how important it is to be fully protected before riding a motorbike. He is known for his expertise on the subject and on other Business and Finance related articles. <br><br>It seems there is still ongoing trail expansion, so I predict a great future for the Gold Canyon trails. You always have to be ready while riding your bike. Buy a bike that has a top quality body made outside of steel, aluminum, carbon fiber or titanium. Hence, having the bicycle that suits you perfectly is much easier now. Re-lube the chain carefully, making sure each link is covered with oil, spin the pedals backwards a few times, then wipe off any excess oil from the chain with an old rag. <br><br>They are usually slightly used and cheaper than a brand new bike. For more information visit: ultimate chopper or email us at chrismartinseo@gmail. The most wanted and suitable bicycle frame for light weight people is made from lighter material. I'd been slowly coming to that conclusion over the course of about a year or so and that mutant nose hair pretty much confirmed it. Scott Motorcycles has been plauged with a popularity of availableness problems on their higher end motorbikes and it looks like 2012 will not be any different. <br><br>No matter what type of bike, no matter how old it is, no matter how much money you'd like to spend. The correct quill stems are sized down with the inside diameter of the fork's steer tube. The best way of getting mountain bike components is shopping online. Mountain bike wheels provide a good mixture of traction, stability, and sturdiness ; and different riding conditions need specific types of mountain bicycle wheels. Some riders advocate the additional technical the trail the greater a hardtail mountain bikeis sought after. |
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| ==Pocklington criterion==
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| The test relies on the '''Pocklington Theorem''' (Pocklington criterion) which is formulated as follows:
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| Let <math> N > 1</math> be an integer, and suppose there exist numbers ''a'' and ''q'' such that
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| '''(1)''' ''q'' is prime, <math>q \vert N - 1</math> and <math>q > \sqrt{N}- 1</math>
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| '''(2)''' <math>a^{N-1} \equiv 1 \pmod{N}</math>
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| '''(3)''' <math>\gcd{(a^{(N-1)/q} - 1 , N)} = 1</math>
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| Then <math>N</math> is prime.<ref name="koblitz">Koblitz, Neal, '''A Course in Number Theory and Cryptography''', 2nd Ed, Springer,1994</ref>
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| ===Proof of this theorem===
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| Suppose ''N'' is not prime. This means there must be a prime ''p'', where <math>p \le \sqrt{N}</math> that divides ''N''.
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| Therefore, <math> q > p - 1 </math> which implies <math>\gcd{(q , p - 1)} = 1</math>.
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| Thus there must exist an integer ''u'' with the property that
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| '''(4)''' <math> uq\equiv 1 \pmod{p - 1}</math>
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| This implies:
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| <math>1 \equiv a^{N-1}\pmod{p}</math>, by '''(2)''' since <math>p \vert N </math>
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| <math> \equiv (a^{N-1})^{u}\equiv a^{u(N-1)} \equiv a^{uq((N-1)/q)}\equiv (a^{uq})^{(N-1)/q}\pmod{p}</math>,
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| <math> \equiv a^{(N-1)/q}\pmod{p}</math>, by '''(4)''' and [[Fermat's little theorem]]
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| This shows the <math>\gcd()</math> of '''(3)''' is actually <math>p</math>, not <math>1</math>; a contradiction.<ref name="koblitz" />
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| The test is simple once the theorem above is established. Given ''N'', seek to find suitable ''a'' and ''q''. If they can be obtained, then ''N'' is prime. Moreover, ''a'' and ''q'' are the certificate of primality. They can be quickly verified to satisfy the conditions of the theorem, confirming ''N'' as prime.
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| A problem which arises is the ability to find a suitable ''q'', that must satisfy (1)–(3) and be provably prime. It is even quite possible that such a ''q'' does not exist. This is a large probability, indeed only 57.8% of the odd primes, ''N'', <math>N \le 10, 000</math> have such a ''q''. To find ''a'' is not nearly so difficult. If ''N'' is prime, and a suitable ''q'' is found, each choice of ''a'' where <math>1 \le a < N</math> will satisfy <math>a^{N-1} \equiv 1\pmod{N}</math>, and so will satisfy (2) as long as ord(''a'') does not divide <math>(N - 1)/q</math>. Thus a randomly chosen ''a'' is likely to work. If ''a'' is a generator mod ''N'' its order is ''N-1'' and so the method is guaranteed to work for this choice.<ref>http://www.mast.queensu.ca/~math418/m418oh/m418og26.pdf</ref>
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| ==Generalized Pocklington method==
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| A generalized version of Pocklington's theorem covers more primes ''N''.
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| '''Corollary:'''
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| Let ''N'' − 1 factor as ''N'' − 1 = ''AB'', where ''A'' and ''B'' are relatively prime, <math> A > \sqrt{N}</math> and the factorization of ''A'' is known.
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| If for every prime factor ''p'' of ''A'' there exists an integer <math>a_p</math> so that
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| : <math>a^{N - 1}_p\equiv 1 \pmod{N}</math>
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| and <math>\gcd{(a^{(N - 1)/p}_p - 1, N)} = 1</math> then ''N'' is prime. The reverse implication also holds: If ''N'' is prime then every prime factor of ''A'' can be written in the above manner.<ref>Blake, Ian F., Seroussi, Gadiel, Smart, Nigel Paul, '''Elliptic Curves in Cryptography''', Cambridge University Press, 1999</ref>
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| '''Proof of Corollary:'''
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| Let ''p'' be a prime dividing ''A'' and let <math>p^e</math> be the maximum power of ''p'' dividing ''A''.
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| Let ''v'' be a prime factor of ''N''. For the <math>a_p</math> from the corollary set
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| <math>b \equiv a^{(N-1)/p^e}_p \pmod{v}</math>. This means
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| <math>b^{p^e} \equiv a^{N-1}_p \equiv 1 \pmod{v}</math> and because of <math>\gcd{(a^{(N-1)/p}_p - 1, N)} = 1</math> also
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| <math>b^{p^{e-1}} \equiv a^{(N-1)/p}_p \not\equiv 1 \pmod{v}</math>.
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| This means that the order of <math>b \pmod{v}</math> is <math>p^e</math>
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| Thus, <math>p^e \vert (v - 1) </math>. The same observation holds for each prime power factor <math>p^e</math> of ''A'',
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| which implies <math>A \vert (v - 1)</math>.
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| Specifically, this means <math>v > A \ge \sqrt{n}.</math>
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| If ''N'' were composite, it would necessarily have a prime factor which is less than or equal to <math>\sqrt{N}</math>. It has been shown that there is no such factor, which implies that ''N'' is prime.
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| To see the converse choose <math>a_p</math> a generator of the integers modulo ''p''.<ref>Washington, Lawrence C., '''Elliptic Curves: Number Theory and Cryptography''', Chapman & Hall/CRC, 2003</ref>
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| ==The test==
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| The Pocklington–Lehmer primality test follows directly from this corollary. We must first partially factor ''N'' − 1 into ''A'' and ''B''. Then we must find an <math>a_p</math> for every prime factor ''p'' of ''A'', which fulfills the conditions of the corollary. If such <math>a_p</math>'s can be found, the Corollary implies that ''N'' is prime.
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| According to Koblitz, <math>a_p</math> = 2 often works.<ref name="koblitz" />
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| ==Example==
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| : <math>N = 11351</math>
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| : <math>N - 1 = 2\cdot 5^2\cdot 227</math>
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| Choose <math>A = 227\cdot5^2</math>, which means <math>B = 2 </math>
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| Now it is clear that <math>\gcd{(A,B)} = 1</math> and <math>A > \sqrt{N}</math>.
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| Next find an <math>a_p</math> for each prime factor ''p'' of ''A''.
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| E.g. choose <math>a_5=2</math>.
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| : <math>a^{N-1}_p \equiv 2^{11350} \equiv 1 \pmod{11351}</math>.
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| : <math>\gcd{(a^{(N-1)/p}_p - 1, N)} = \gcd{(2^{2\cdot 5\cdot 227} - 1, 11351)} = 1.</math>
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| So <math>a_5=2</math> satisfies the necessary conditions. Choose <math>a_{227} = 7</math>.
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| : <math>a^{N-1}_p \equiv 7^{11350} \equiv 1 \pmod{11351}</math>
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| and
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| : <math>\gcd{(a^{(N-1)/p}_p - 1, N)} = \gcd(7^{2\cdot 25} - 1, 11351) = 1.</math>
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| So both <math>a_p</math>'s work and thus ''N'' is prime.
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| We have chosen a small prime for calculation purposes but in practice when we start factoring ''A'' we will get factors that themselves must be checked for primality. It is not a proof of primality until we know our factors of ''A'' are prime as well. If we get a factor of ''A'' where primality is not certain, the test must be performed on this factor as well. This gives rise to a so-called down-run procedure, where the primality of a number is evaluated via the primality of a series of smaller numbers.
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| In our case, we can say with certainty that 2, 5, and 227 are prime, and thus we have proved our result. The certificate in our case is the list of <math>a_p</math>'s, which can quickly be checked in the corollary.
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| If our example had given rise to a down-run sequence, the certificate would be more complicated. It would first consist of our initial round of <math>a_p</math>'s which correspond to the 'prime' factors of ''A''; Next, for the factor(s) of ''A'' of which primality was uncertain, we would have more <math>a_p</math>'s, and so on for factors of these factors until we reach factors of which primality is certain. This can continue for many layers if the initial prime is large, but the important thing to note, is that a simple certificate can be produced, containing at each level the prime to be tested, and the corresponding <math>a_p</math>'s, which can easily be verified. If at any level we cannot find <math>a_p</math>'s then we cannot say that ''N'' is prime.
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| The biggest difficulty with this test is the necessity of discovering prime factors of ''N - 1'', in essence, factoring ''N'' − 1. In practice this could be extremely difficult. Finding <math>a_p</math>'s is a less difficult problem.<ref>{{cite book|authors=Roberto Avanzi, Henri Cohen, Christophe Doche, Gerhard Frey, Tanja Lange, Kim Nguyen, Frederik Vercauteren|title=Handbook of Elliptic and Hyperelliptic Curve Cryptography|publisher=Chapman & Hall/CRC|location=Boca Raton|year=2005|url=http://www.hyperelliptic.org/HEHCC}}</ref>
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| == References ==
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| <!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
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| {{Reflist}}
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| == External links ==
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| {{Number-theoretic algorithms}}
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| <!--- Categories --->
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| {{DEFAULTSORT:Pocklington Primality Test}}
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| [[Category:Primality tests]]
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Make sure you can hear your breathe (except in public places where you may want to be more discreet) and you can see the stomach moving as this will help you to build up a rhythm. A little more than 30 miles, with a 5,200 foot elevation gain, and you'll ride over the finish line. In case you adored this informative article and also you want to obtain more details with regards to Popular mountain bike sizing. i implore you to stop by the page. The contemporary bikes for mountains are provided with stronger and lighter frame types in addition to pioneering form and design. There are more than 15 circuits in the sport of professional mountain biking. If you've ridden for years and years and know what you need, buying online does make sense.
As hardtails have front-only suspension, they have less pedal bob and increased pedal stroke efficiency when compared to dual suspension MTB bikes. You can install an electric motor on your regular bike in about an hour or hire a bike mechanic to do it. An alternative to Clipless pedals are Mountain Bike shoes with Shimano Pedaling Dynamics (SPD). He or she doesn't have to get into an accident just to understand how important it is to be fully protected before riding a motorbike. He is known for his expertise on the subject and on other Business and Finance related articles.
It seems there is still ongoing trail expansion, so I predict a great future for the Gold Canyon trails. You always have to be ready while riding your bike. Buy a bike that has a top quality body made outside of steel, aluminum, carbon fiber or titanium. Hence, having the bicycle that suits you perfectly is much easier now. Re-lube the chain carefully, making sure each link is covered with oil, spin the pedals backwards a few times, then wipe off any excess oil from the chain with an old rag.
They are usually slightly used and cheaper than a brand new bike. For more information visit: ultimate chopper or email us at chrismartinseo@gmail. The most wanted and suitable bicycle frame for light weight people is made from lighter material. I'd been slowly coming to that conclusion over the course of about a year or so and that mutant nose hair pretty much confirmed it. Scott Motorcycles has been plauged with a popularity of availableness problems on their higher end motorbikes and it looks like 2012 will not be any different.
No matter what type of bike, no matter how old it is, no matter how much money you'd like to spend. The correct quill stems are sized down with the inside diameter of the fork's steer tube. The best way of getting mountain bike components is shopping online. Mountain bike wheels provide a good mixture of traction, stability, and sturdiness ; and different riding conditions need specific types of mountain bicycle wheels. Some riders advocate the additional technical the trail the greater a hardtail mountain bikeis sought after.