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	PlanetMath: probable prime
		believed to be a prime number because it has passed some preliminary primality test relative to a given base, or a pattern suggests it might be prime, but it has not yet been subjected to a conclusive primality test.
		Once a probable prime is conclusively shown to be a prime, it of course loses the label "probable." It also loses it if conclusively shown to be composite, but in that case it might then be called a pseudoprime relative to base
		This is version 1 of probable prime, born on 2006-05-04.
	planetmath.org /encyclopedia/ProbablePrime.html (148 words)

	Prime number - ExampleProblems.com
		The prime number theorem says that the proportion of primes less than x is asymptotic to 1/ln x (in other words, as x gets very large, the likelihood that a number less than x is prime is inversely proportional to the number of digits in x).
		A probable prime is an integer which, by virtue of having passed a certain test, is considered to be probably prime.
		With this definition, the primes of the field Q of rational numbers are represented by the standard absolute value function (known as the "infinite prime") as well as by the p-adic valuations on Q, for every prime number p.
	www.exampleproblems.com /wiki/index.php/Prime_number (3311 words)

	PlanetMath: examples of probable primes
		is not a probable prime, because even though it is much larger than either of the probable primes given above, it is clearly divisible by
		Cross-references: divisible, even, average, Mersenne prime, period, division, square root, base, probable prime
		This is version 2 of examples of probable primes, born on 2006-05-04, modified 2006-11-03.
	planetmath.org /encyclopedia/ExamplesOfProbablePrimes.html (158 words)

	Probable prime - Wikipedia, the free encyclopedia
		While there may be probable primes that are composite (called pseudoprimes), the condition is generally chosen in order to make such exceptions rare.
		A strong probable prime to base a is called a strong pseudoprime to base a.
		Probable primality is a basis for efficient primality testing algorithms, which find application in cryptography.
	en.wikipedia.org /wiki/Probable_prime (493 words)

	Math Games: Prime Megagap
		An example of a prime gap of size 20 is found between primes 887 and 907.
		Probable primes are prime with a 99.99999999999999% certainty, and this battery of tests never fails for numbers under 10^16.
		A list of currently unverifiable probable primes is maintained at primenumbers.net.
	www.maa.org /editorial/mathgames/mathgames_01_25_04.html (862 words)

	The Prime Glossary: probable prime
		Some of these probable primes are actually composites, we call these numbers pseudoprimes.
		There are only 1,092,19,258 base two probable primes less than 25,000,000,000; but only 21,853 of these are pseudoprimes, so the probable prime test base two would fail only 0.0000874% of the time in this region.
		Below are the odd composite probable primes (pseudoprimes) less than 500 for bases 2, 3,..., 20.
	primes.utm.edu /glossary/page.php?sort=PRP (310 words)

	Unique prime - Wikipedia, the free encyclopedia
		A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equivalent to the period length of the reciprocal of q, 1 / q.
		At present, more than fifty unique primes or probable primes are known.
		Though they are rare, based on the occurrence of repunit primes and probable primes, it is conjectured strongly that there are infinitely many unique primes.
	en.wikipedia.org /wiki/Unique_prime (273 words)

	質數的測試方法
		is prime by theorem 2 with a = 3.
		prime number theorem the probability that a random positive integer k is prime is (asymptotic to) 1/log(k) (where log x is the natural logarithm).
		In particular, the probability density f(t) for the gaps and the probability p(t) the gap length is as follows [Ross93, p215].
	www.freewebtown.com /wolfram_lin/marsenne/primality.html (8166 words)

	MATHEWS: Circular, Permutable, Truncatable and Deletable Primes (Site not responding. Last check: 2007-10-13)
		Therefore 187 is a circular prime because all cyclic rotations are prime.
		Permutable primes are primes with at least two distinct digits which remain prime when permuting the digits.
		Left-truncatable primes p of length n with the additional property that no prime with length n+1 can have its leftmost digit removed to produce p are called Henry VIII primes (Kahan and Weintraub, 1998).
	www.wschnei.de /digit-related-numbers/circular-primes.html (577 words)

	Supplement to "palindromic prime pyramids"
		In the paper "Palindromic prime pyramids" [HC2000] we discuss a type of pyramid first proposed by G. Honaker, Jr.
		Note that each row is a palindromic prime (a palprime) with the previous row as the central digit.
		For testing the primality of the numbers of the pyramid I used a "strong-pseudo prime" test and code, according to the pseudocode (Algorithm 6.1) described in the David M. Bressoud's book [Bressoud89] "Factorization and primality testing", Springer-Verlag, p.
	www.utm.edu /staff/caldwell/supplements (346 words)

	Testing for Prime Repunits (Site not responding. Last check: 2007-10-13)
		This just means that they are "probably" primes, because composites that satisfy this test are extremely rare.
		For a long time those five numbers were the only known prime repunits (in decimal), but in September 1999 Harvey Dubner found the next "probable prime" repunit with p = 49081.
		This most likely is a prime, but it would be prohibitive (by any known methods) to give a deterministic proof.
	www.mathpages.com /home/kmath317.htm (210 words)

	Primes and Primality Testing
		A probable prime is an integer that has failed some compositeness test; if an integer passes a compositeness test it will be composite, but there is a (small) probability that a composite number will fail the test and is hence called a probable prime.
		Each Miller-Rabin test for instance, has a probability of less than 1/4 of declaring a composite number probably prime; in practice that means that numbers that fail several such cheap independent Miller-Rabin compositeness tests will be prime.
		In such cases n is called probably prime of order k, and in some sense the probability that n is composite is less than 4^(-k).
	www.umich.edu /~gpcc/scs/magma/text539.htm (1059 words)

	Puzzle 126. Some conspicuous probable primes
		This requires that at least one of the probable primes asked has less than 4000 digits and more than 3106 digits.
		If all of these 3 probable primes are too larger than 4000 digits, then the task will have to be solved in the near?
		If all the asked probable primes are out of the suggested range maybe you would be so kind to submit another digital-conspicuous probable prime and candidate to be the record for the Titanix code, with a quantity of digits in the proper range.
	www.primepuzzles.net /puzzles/puzz_126.htm (240 words)

	PrimeForm for Windows
		While probable primality does not in itself prove primality, it is a valuable tool and a very quick test to establish if a number is composite.
		If probable primality is established, it may be possible to prove primality by further testing.
		Probable Prime mode will be the mode you use the most.
	pages.prodigy.net /chris_nash/primeform.html (1814 words)

	MATHEWS: Prime Numbers and Digit Related Topics (Site not responding. Last check: 2007-10-13)
		In this paper we are interested in prime numbers as a special case of digit related numbers.
		Left-truncatable primes p of length n with the additional property that no prime with length n+1 can have its leftmost digit removed to produce p are called Henry VIII primes.
		Dihedral primes are defined by Mike Keith as integers n which displayed on a calulator have the property that n, n upside down, n in a mirror and n upside down and in a mirror are all primes.
	www.wschnei.de /digit-related-numbers/primes.html (892 words)

	Pseudoprimes/Probable Primes
		The proliferation of probable prime tests in recent years has produced a plethora of definitions with the word "pseudoprime" in them.
		It is an application of ideas in the first paper to produce a probable prime test that has an expected running time 3 times as long as that of the Strong Probable Prime Test, but is more than 3 times as accurate.
		I strongly believe that some sub-product of these primes is a Carmichael number and a Lucas pseudoprime for the Fibonacci sequence, and also is 2 or 3 mod 5.
	www.pseudoprime.com /pseudo.html (357 words)

	Generating Prime Numbers and When Not to Overload Methods
		A prime number is an integer greater than 1, with only itself and 1 as divisors.
		The generated numbers are probably prime, but there is a slight chance that they might not be.
		The standard set by ANSI X9.80 specifies a certainty for use in private-key encryption: the probability that a generated prime is actually prime is greater than 1 minus 1/2 to the 100th power.
	java.sun.com /developer/JDCTechTips/2002/tt0806.html (1653 words)

	Frobenius Probable Prime Test (Site not responding. Last check: 2007-10-13)
		I wrote an implementation of the rqf test at the mpz level, and in pratice it was faster (than the strong probable prime test) from about 2 to 4 limbs for a error of 1 in 10^10.
		You tweek the theorem so that any number that passes is also a strong probable prime and then do this part of the test first.
		I kind of in the process of mpn'ifing it, but it will take some time because all the underlying functions need to mpn'ified I can send you the code if you want, but I warn you, It's not neat or commented, it should be correct though....
	www.swox.com /list-archives/gmp-devel/2003-May/000126.html (307 words)

	Puzzle 20.- Reversible Primes
		13 is the least reversible non palindrome prime because 31 is also a prime.
		Reading one more time the Rudolph Ondrejka Primes Collection I saw recently that there is a record for this kind of primes established by H.
		In each case the decimal prime and the reverse is k and reverse(k) with 997 9's on both sides.
	www.primepuzzles.net /puzzles/puzz_020.htm (317 words)

	The Prime Glossary: strong probable prime
		All integers n greater than one which fail this test are composite; integers that pass it might be prime.
		A test based on these results is quite fast, especially when combined with trial division by the first few primes.
		Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics volume 126, Birkhäuser Boston, Boston, MA, ISBN 0-8176-3743-5.
	primes.utm.edu /glossary/page.php?sort=StrongPRP (768 words)

	Part 2: Primes Related to Corollary 7.1
		The next (probable) prime, found by H. Zeisel, corresponds to k = 1885, and for a long time this was the largest value of k known to yield a prime (or probable prime).
		+ k is a probable prime with k = 51381.
		The values of q for any given prime p represent a closed set (mod p) of powers of 2 times one particular base.
	www.mathpages.com /home/kmath006/part2/part2.htm (1586 words)

	Factorization table
		So, p85 means a prime number of 85 digits, and c104 means a composite number (whose factors are unknown) of 104 digits.
		Note that some of the larger prime factors are only “probable primes”, and not proven primes.
		In this table, the only “concatenated prime numbers” which are prime are 2, 23, and 2357.
	home.netcom.com /~jrhowell/math/primetbl.htm (278 words)

	Lucas and Perrin Probablistic Prime Tests
		All true primes are declared prime but some composites are also declared prime so the mathematical filter is not perfect.
		Though the probability of a composite integer being declared prime by the ISPRIME() function is extremely low, examples of such integers are known.
		Don't confuse this simple Lucas probable prime test with the more comprehensive Lucas-Lehmer test that is based on the same principles.
	www.lrbcg.com /jtCullen/Math4.htm (1312 words)

	Amazon.com: Prime Numbers: Books: Richard Crandall,Carl B. Pomerance (Site not responding. Last check: 2007-10-13)
		Primes are at the heart of PKI and its RSA implementation.
		While recently published, this book is shaping to become the standard reference on the theory that surrounds prime numbers in a computational setting, drawing from all branches of number theory, as well as abstract algebra, analysis, combinatorics, statistics, complexity theory and elliptic curves.
		The first chapter is an overview of theoretical and computational developments, with anything from Euclid's proof of the infinitude of primes, Riemann's study of the zeta function, down to the latest huge computation of the twin prime constant and zeros of the zeta function in the critical line.
	www.amazon.com /Prime-Numbers-Richard-Crandall/dp/0387947779 (1682 words)

	[No title]
		f is 1 to indicate that n is a probable prime, 0 for composite, and -1 if n is a square.
		It is NOT assumed that the moduli M[i] are pairwise relatively prime.
		The prime factors and their multiplicities are stored in P[] and M[] respectively.
	archives.math.utk.edu /software/msdos/number.theory/malm/malm.readme (2415 words)

	Illegal Prime Number?
		I have been a large prime prover (parse that either way!) for quite a while now (I refer you to, for example, the "Near Repdigit" record on Professor Caldwell's Prime Pages), and I tried to think how I could come up with a number archivable on those pages.
		The Prime Pages is the world's definitive resource for large prime number archiving and information; Professor Caldwell, who manages it, has put an awfully large amount of effort into deveoloping those pages, and into administering the archival of large primes, and has put a lot of thought into the criteria for their archivability.
		However it has become "interesting" enough to be one of the curios on the Prime Curios site, which is part of the Prime Pages site, and is for the more "fun" aspects of prime numbers.
	asdf.org /~fatphil/maths/illegal1.html (1435 words)

	premiers.html
		3a(7,4)/a(6,4)/379 = 147419592855542737789440527865055084545487095492014413250889447284402837791 12958872916610421681465870692267621675809786507485857789764050033536903097 (c149) by ecm a prime factor of 28 digits: 1367919568960169498955547991 and a composite cofactor of 122 digits: 10776919652345088930014977534732438432923444510241640662127136919959539939 309242894484157283897657224612382419685688934767.
		Found probable prime factor of 14 digits: 57675946866011.
		Probable prime cofactor 5453737259593603153452780353429669215977523202978062215080688974659422772564617248370990659731 has 94 digits c187:=7546313245332712339298521326282679343981074803446066121383958122243000176722655909418568161596015608619696047411218156801295218739286189139913118666146671830007153213494514895233801796101 Found probable prime factor of 10 digits: 7414936579
	igd.univ-lyon1.fr /~mizony/premiers.html (1266 words)

	The Repdigit Prime Problems
		If you found a probable prime then tell me right away and you can stop your search immidiately for that specific n.
		A probable prime is very likely to be a prime but there is still a chance that A is a composite number.
		It is conjectured that only primes can pass this test, if not, the chance a composite number can pass this test is much much smaller than 10^-15 and since we are working with probable primes, it's very likely that none of them is composite.
	www.freewebs.com /dries5/prime_problems.htm (1136 words)

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