Where results make sense

About us   |   Why use us?   |   Reviews   |   PR   |   Contact us

Euler-Jacobi pseudoprime - Wikipedia, the free encyclopedia Every Euler-Jacobi pseudoprime is also a Fermat pseudoprime and an Euler pseudoprime. There are no numbers which are Euler-Jacobi pseudoprimes to all bases as Carmichael numbers are. The table below gives all Euler-Jacobi pseudoprimes less than 10000 for some prime bases a, this table is in the process of being checked and should be used with caution until this notice is removed. en.wikipedia.org /wiki/Euler-Jacobi_pseudoprime   (434 words)

Euler pseudoprime - Wikipedia, the free encyclopedia It is not possible to produce a definite test of primality based on whether a number is an Euler pseudoprime because there exist absolute Euler pseudoprimes, numbers which are Euler pseudoprimes to every base relatively prime to themselves. The absolute Euler pseudoprimes are a subset of the absolute Fermat pseudoprimes, or Carmichael numbers, and the smallest absolute Euler pseudoprime is 1729 = 7·13·19. = (a/n) (mod n), where (a,n)=1 and (a/n) is the Jacobi symbol, is sometimes used for a definition of an Euler pseudoprime. en.wikipedia.org /wiki/Euler_pseudoprime   (293 words)

Number Theory Glossary A Carmichael Number is a composite number which passes the Fermat pseudoprime test for all bases. Also called the Euler pseudoprime test, this test was originally proposed by Solovay and Strassen in SIAM J. Computing, 6 (1977), 84-85 and 7 (1978), 118. If an integer is a strong pseudoprime it is also a Fermat pseudoprime and an Euler pseudoprime. www.math.umbc.edu /~campbell/NumbThy/Class/Glossary.html   (827 words)

PSEUDOPRIME   (Site not responding. Last check: 2007-09-17) This term is an esoteric pun derived from number theory: a number that passes a certain kind of "primality test" may be called a `pseudoprime' (all primes pass any such test, but so do some composite numbers), and any number that passes several is, in some sense, almost certainly prime. The hacker backgammon usage stems from the idea that a pseudoprime is almost as good as a prime: it will do the same job unless you are unlucky. Pseudoprimes to base 2 are called Poulet numbers or sometimes Sarrus numbers or Fermatians (SIDN A001567). www.websters-online-dictionary.org /ps/pseudoprime.html   (724 words)

Math 5410 Midterm Exam (b) Prove that any Euler pseudoprime to the base b is a pseudoprime to the base b. - 1 is a strong pseudoprime and an Euler pseudoprime to the base 2. This is a cyclic shift cipher by 13 (i.e. www-math.cudenver.edu /~wcherowi/courses/m5410/m5410mid.html   (969 words)

Fermat number Summary Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. It is widely believed that Fermat was aware of Euler's result, so it seems curious why he failed to follow through on the straightforward calculation to find the factor. One common explanation is that Fermat made a computational mistake and was so convinced of the correctness of his claim that he failed to double-check his work. www.bookrags.com /Fermat_number   (1320 words)

Euler pseudoprime . Fermat's little theorem . Euler-Jacobi pseudoprime . Probable prime . Carmichael number   (Site not responding. Last check: 2007-09-17) An Odd number odd composite number composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and : a^ \equiv \pm 1\pmod where mod refers to the modular arithmetic modulo operation. If a and p are coprime numbers such that a p −1 − 1 is divisible by p, then p need not be prime. If it is not, then p is called a pseudoprime to base a. www.uk.fraquisanto.net /Euler_pseudoprime   (500 words)

Euler-Jacobi pseudoprime   (Site not responding. Last check: 2007-09-17) The motivation for this definition is the that all prime numbers n satisfy the above equation as explained the Legendre symbol article. There are no numbers which are pseudoprimes to all bases as Carmichael numbers are. The table below gives all Euler-Jacobi pseudoprimes than 10000 for some prime bases a this table is in the process being checked and should be used with until this notice is removed. www.freeglossary.com /Euler-Jacobi_pseudoprime   (336 words)

PlanetMath: Miller-Rabin prime test Note that this theorem states that there are no such things as Carmichael numbers for strong pseudoprimes (i.e. There is no need to calculate the Jacobi symbol. Cross-references: even, witnesses, Jacobi symbol, calculate, Solovay-Strassen test, order, divisor, Carmichael numbers, basis, composite number, Euler's phi-function, prime, odd, odd number planetmath.org /encyclopedia/MillerRabinPrimeTest.html   (207 words)

Encyclopedia Search pseudoprime An odd composite integer n is...is called an Euler- pseudoprime to base a, if a and n are coprime, and a (n -1)/2 = (a /... symbol, is sometimes used for a definition of an Euler pseudoprime. www.encyclopedian.com /search.php?searWords=Jacobi   (133 words)

Section 7 For the two types of pseudoprime already mentioned, there is no measure of the probability of composite n surviving repeated tests; we just accept that it decreases substantially at each iteration. Every strong pseudoprime to the base a is already an Euler pseudoprime to the base a. It is better to use the repeat pseudoprime method for a smaller range of test values, and using the fact that the probability of a composite n passing each iteration of the test is at most 1/4, so that after k successful tests, the probability that n is prime is greater than 1 www.glasgowg43.freeserve.co.uk /pfaq7.htm   (2603 words)

[No title] Carmichael numbers +------------------------------------------------------------ Carmichael numbers are natural numbers which are Fermat pseudoprime to any base. Jacobi symbol +------------------------------------------------------------ Jacobi symbol If m=p_1...p_k is the prime factorization of m, define Jacobi(n,m) as the product of Legendre(n,p_i), where Legendre(n,p) denotes the Legendre symbol of n and p and m=p_1.s p_k is the prime factorization of m. Jacobi symbol +------------------------------------------------------------ A positive integer a which generates the multiplicative group modulo a prime number p is called a primitive root of p. www.math.harvard.edu /~knill/sofia/data/number.txt   (1516 words)

Euler's criterion . Leonhard Euler . Euler-Jacobi pseudoprime . Legendre symbol . Fermat's little theorem Daniel Bernoulli, established the law that the torque on a thin elastic beam is proportional to a measure of the elastic elasticity of the material and the moment of... odd number odd composite number composite integer n is called an Euler-Jacobi pseudoprime to base a, if a and n are coprime, and : a n − 1 2 = a n modular arithmetic mod n, where a n is the Jacobi symbol. The motivation for this definition is the fact that all prime numbers n satisfy... www.uk.kunsimuna.net /Euler%27s_criterion_UK_206022_mu   (364 words)

Andrzej Rotkiewicz In my book Pseudoprime Numbers and Their Generalizations I gave all what was known about pseudoprimes up to 1972. Pseudoprime numbers and their generalizations, Student Association of the Faculty of Sciences, University of Novi Sad, Novi Sad 1972, pp. On the pseudoprimes with respect to the Lucas sequences, Bull. www.impan.gov.pl /User/rotkiewi   (488 words)

[No title]   (Site not responding. Last check: 2007-09-17) Another step up is taken with a definition of the Jacobi symbol and its properties, which we need later in the chapter. Once the concept of strong pseudoprimes, liars, and witnesses are introduced, we are in a position to present the Miller-Rabin-Selfridge strong pseudoprime. A brief history of how the work of Legendre, Euler, Kraitchik and Lehmer led to the development of these algorithms is also provided, as is a discussion of how the notions can be generalized. www.math.ucalgary.ca /~ramollin/cryptopref.html   (2293 words)

Amazon.com: "Euler's Criterion": Key Phrase page   (Site not responding. Last check: 2007-09-17) See all pages with references to Euler's Criterion. EULER'S CRITERION FORMULATED The change of the base from 2 to 3 changes the divisibility laws from Eqs. For future reference, we record the following well-known result, which is known as Euler's Criterion. www.amazon.com /phrase/Euler's-Criterion   (557 words)

SOAR Spring 2003 Course (All About Numbers) On the other hand, a Lucas pseudoprime is a composite number that is not caught by this Lucas test. We checked some numbers by hand and saw that 341 is a base 2 pseudoprime, 121 is a base 3 pseudoprime, 217 is a base 5 pseudoprime, and 373 is, in fact, simply prime. A number n that is a base b pseudoprime for every base b relatively prime to n is called a Carmichael number. www.math.toronto.edu /mathnet/SOAR2003/Spring   (3611 words)

Amazon.com: "strong pseudoprimes": Key Phrase page   (Site not responding. Last check: 2007-09-17) If this technique is to be carried much further, we have to study what is called strong pseudoprimes rather than using the ordinary Fermat pseudoprimes, 86... Strong pseudoprimes base 2 are referred to simply as strong pseudoprimes. Unfortunately for every base b there are infinitely many strong pseudoprimes which are not primes, but since strong pseudoprimes are pseudoprimes these strong pseudoprimes are again very rare. www.amazon.com /phrase/strong-pseudoprimes   (525 words)

The Prime Glossary: Euler probable prime is one modulo p.  Euler was able to prove the stronger statement: a If this is the case and n is composite, then we say n is an Euler pseudoprime (base a). Below is a table of the odd composite Euler PRP's less than 3000 and the percentage of the odd composites in this range that are not exposed as composite by this test. primes.utm.edu /glossary/page.php?sort=EulerPRP   (240 words)

Mathematica Usage Messages EulerGamma is Euler's constant gamma, with numerical value 0.577216.... It is the number of positive integers less than n which are relatively prime to n. is the logarithm of the Euler gamma function Gamma(z). vlado.fmf.uni-lj.si /vlado/symbol/usage.htm   (11884 words)

Number theory (Here we used the same notation [a/b;] for the Legendre and the Jacobi symbols; this is confusing but seems to be the current practice.) The Jacobi symbol is equal to 0 if m, n are not mutually prime (have a common factor). The Jacobi symbol and the Legendre symbol have values +1, -1 or 0. The Jacobi symbol can be efficiently computed without knowing the full factorization of the number n. mathsrv.ku-eichstaett.de /MGF/homes/grothmann/euler/yacas/refchapter4.html   (1987 words)

BigInteger Members   (Site not responding. Last check: 2007-09-17) The two pseudoprimes p and q are fixed, but the two RSA keys are generated for each round of testing. True if "this" is a strong pseudoprime to randomly chosen bases. Probabilistic prime test based on Solovay-Strassen (Euler Criterion).True if "this" is a Euler pseudoprime to randomly chosen bases. www.pivo.com /doc/lib/Pivo.SSH.BigIntegerMembers.html   (444 words)

DESCRIPTION OF CALC This finds the first p, p=b(mod a), m ≤ p, which passes the strong base 2 pseudoprime test and the Lucas pseudoprime test. n is subjected to a strong pseudoprime test to base 2, together with a Lucas pseudoprime test. If z=0 is returned, n is composite, while if z=1 is returned, then n is a Lucas probable prime, as well as a base 2 strong pseudoprime. www.numbertheory.org /calc/krm_calc.html   (4466 words)

pseudoprime - OneLook Dictionary Search Tip: Click on the first link on a line below to go directly to a page where "pseudoprime" is defined. Pseudoprime : Eric Weisstein's World of Mathematics [home, info] Phrases that include pseudoprime: frobenius pseudoprime, perrin pseudoprime, euler jacobi pseudoprime, fibonacci pseudoprime, somer-lucas pseudoprime, more... www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=pseudoprime   (102 words)

Euler-Jacobi pseudoprime . Legendre symbol . Fermat's little theorem In number theory, an odd number odd composite number composite integer n is called an Euler-Jacobi pseudoprime to base a, if a and n are coprime, and : a n − 1 2 = a n modular arithmetic mod n, where a n is the Jacobi symbol. These tests are over twice as strong as tests based on Fermat s little theorem. There are no numbers which are Euler-Jacobi pseudoprimes to all bases as www.uk.kunsimuna.net /Euler-Jacobi_pseudoprime_UK_973799_hp   (328 words)

Euler-Jacobi pseudoprime Solovay and Strassen showed that for every composite n, for at least n/2 bases less than n, n isn't an Euler-Jacobi pseudoprime. The table below gives all Euler-Jacobi pseudoprimes less than 10000 for some prime bases a, this table is in the process of being checked and should be used with caution until thisn'tice is removed. A considerable force of Italian the Ruhr, and apparently out of convenient supporting distance from upon this tender spot, break through the lines, and bring on a general force, he divided them into two formidable bands, one under the charge. www.freearchive.info /eu/euler-jacobi-pseudoprime.html   (632 words)

質數的測試方法 Some early articles call all numbers satisfying this test pseudoprimes, but now the term pseudoprime is properly reserved for composite probable-primes. Euler PRP's, but we wish to develop a stronger test here.) If (n-1)/2 is even, we can easily take another square root... It has been proven ([Monier80] and [Rabin80]) that the strong probable primality test is wrong no more than 1/4th of the time (3 out of 4 numbers which pass it will be prime). www.freewebtown.com /wolfram_lin/marsenne/primality.html   (8166 words)

The Yacas Book of Algorithms Composite strongly-probably-prime numbers for base b are called strong pseudoprimes for base b. computations have shown that there are no strong pseudoprimes simultaneously for bases 2, 3, 5, 7, 11, 13 and 17. Using these identities, we can recursively reduce the computation of the Jacobi symbol [a/b;] to the computation of the Jacobi symbol for numbers that are on the average half as large. mathsrv.ku-eichstaett.de /MGF/homes/grothmann/euler/yacas/Algo.html   (15814 words)

How to find primes and prove primality (merged version)   (Site not responding. Last check: 2007-09-17) There are 1,091,987,405 primes less than 25,000,000,000; but only 21,853 pseudoprimes base two [PSW80], so Henri Cohen joked that 2-PRP's are "industrial grade primes" [Pomerance84, p5]. There may be relatively few pseudoprimes, but there are still infinitely many of them for every base a>1, so we need a tougher test. Jon Grantham's "Frobenius pseudoprimes" can be used to create a test (see [Grantham98]) that takes three times as long as the SPRP test, but is far more than three times as strong (the error rate is less than 1/7710). primes.utm.edu /prove/merged.html   (6842 words)

Try your search on: Qwika (all wikis)

About us   |   Why use us?   |   Reviews   |   Press   |   Contact us   Copyright © 2005-2007 www.factbites.com Usage implies agreement with terms.