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  • What use are prime numbers anyway?
    ==So it would seem to be a fair question "What use are prime numbers?"== ...e would not exist as we know it, and all of this security depends on prime numbers.
    3 KB (497 words) - 07:17, 22 May 2020
  • Strong law of small numbers
    ...athematician]] Richard K. Guy published a paper ''"The Strong Law of Small Numbers"''. In it he states, :"There aren't enough small numbers to meet the many demands made of them."
    1 KB (197 words) - 15:02, 11 February 2019

Page text matches

  • Williams prime
    ...both signs, there're four different types of numbers similiar to Williams numbers.
    5 KB (744 words) - 07:30, 5 August 2019
  • Great Internet Mersenne Prime Search
    ...be downloaded from the Internet, in order to search for [[Mersenne prime]] numbers. ...an, who was born in 1588. Mersenne investigated a particular type of prime numbers: 2<sup>p</sup> - 1, in which ''p'' is an ordinary [[prime]].
    3 KB (450 words) - 14:37, 21 August 2019
  • List of known Mersenne primes
    *[https://www.mersenne.org/primes/ List of known Mersenne prime numbers] at Mersenne.org *[http://www.utm.edu/research/primes/mersenne.shtml prime Mersenne Numbers - History, Theorems and Lists] Explanation
    2 KB (360 words) - 09:44, 6 March 2019
  • Mersenne prime
    ...Mersenne number]]s (not necessarily primes, but candidates for primes) are numbers that are one less than a power of two; hence, ...umber|even]] perfect numbers have this form. No [[odd number|odd]] perfect numbers are known, and it is suspected that none exists.
    5 KB (857 words) - 14:53, 19 September 2021
  • Mersenne number
    ==Properties of Mersenne numbers== Mersenne numbers share several properties:
    2 KB (351 words) - 11:28, 7 March 2019
  • Leonhard Euler
    ...term 'function' in this context. He is the only mathematician to have two numbers named after him.
    16 KB (2,614 words) - 11:48, 14 January 2024
  • Pierre de Fermat
    429 bytes (63 words) - 11:44, 14 January 2024
  • Landon Curt Noll
    2 KB (333 words) - 12:40, 9 February 2022
  • George Woltman
    ...ing project|distributed computing project]] researching [[Mersenne prime]] numbers using his software [[Prime95]] and [[Prime95|MPrime]]. He graduated from th
    1 KB (164 words) - 14:40, 21 August 2019
  • Fermat number
    where {{Vn}} is a [[non-negative]] [[integer]]. The first eight Fermat numbers are (see {{OEIS|l|A000215}}): ...e found at [http://www.prothsearch.com/fermat.html Prime Factors of Fermat Numbers]
    12 KB (1,913 words) - 14:35, 9 August 2021
  • Derrick Henry Lehmer
    ...stencils. In the days before computers [[Factorization|factorising]] large numbers was a laborious task and many methods had been tried to make it easier. [[F ...ciently influential that the terms in this sequence are now called 'Lehmer Numbers'. He also clarified and extended Lucas' use of the Fermat congruence in pri
    6 KB (1,033 words) - 01:13, 15 January 2024
  • Megaprime
    ...are infinitely primes. In fact, since there are only finitely many natural numbers with less than {{Num|1000000}} digits, "nearly all" primes are megaprimes.
    806 bytes (111 words) - 07:59, 14 July 2021
  • Gigaprime
    ...are infinitely primes. In fact, since there are only finitely many natural numbers with less than {{Num|1000000000}} digits, "nearly all" primes are gigaprime
    871 bytes (119 words) - 07:54, 14 July 2021
  • Cunningham project
    ...the supply of numbers to be factored is low, the project starts factoring numbers with higher exponents, tracking the advances in factorization algorithms an For Mersenne numbers of the form <math>2^n-1</math>, even this trivial factor is not possible fo
    7 KB (1,150 words) - 23:48, 19 April 2023
  • 2 Minus Tables
    ==Factorizations Of Cunningham Numbers C<sup>-</sup>(2,n) = 2<sup>n</sup> - 1==
    2 KB (176 words) - 12:01, 13 February 2019
  • Lucas-Lehmer test
    ...an [[Édouard Lucas]] (1842 - 91) developed an entirely new way of proving numbers prime without attempting to find all of their factors. Instead, he showed t ...ger number, the Lucas-Lehmer number, is calculated as one in a sequence of numbers where each number is the previous number squared, minus 2. So that where S<
    20 KB (3,572 words) - 14:30, 17 February 2019
  • Édouard Lucas
    ...] is named after him. He devised a new method for testing the primality of numbers that did not require finding all of their factors. In 1930, [[Derrick Henry
    2 KB (296 words) - 01:09, 15 January 2024
  • Edson Smith
    ...e people, sort of a passion. There's really no guarantee that any of these numbers exist. We don't know they're there until we find them. So it's exciting to
    4 KB (564 words) - 00:11, 15 January 2024
  • Digit
    ...r "7") used in numerals (combinations of symbols, e.g. "37"), to represent numbers, ([[integer]]s or [[real number]]s) in positional numeral systems. The name
    1 KB (171 words) - 10:17, 18 January 2019
  • Decimal
    ...d the radix point) that is sometimes used to separate the positions of the numbers in this system. This is the common every-day numbering system that people u ...han ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers, no matter how large. These digits are often used with a decimal separator
    1 KB (190 words) - 10:23, 18 January 2019
  • Base
    ...number of different [[digit]]s that a system of counting uses to represent numbers. For example, the most commonly used base today is the decimal system. Beca ==Numbers in different bases==
    2 KB (399 words) - 10:37, 18 January 2019
  • Whole number
    413 bytes (54 words) - 09:51, 8 February 2019
  • Integer
    ...fer only to the positive integers (with or without zero). Like the natural numbers, the integers form a countably infinite set. ...ative natural numbers, and, importantly, zero, '''Z''' (unlike the natural numbers) is also closed under [[subtraction]]. '''Z''' is not closed under the oper
    3 KB (404 words) - 14:58, 26 March 2023
  • Mathematics
    :*[[Arithmetic]] - The study of whole numbers and fractions. ...Algebra]] - The use of abstract symbols to represent mathematical objects (numbers, lines, matrices, transformations), and the study of the rules for combinin
    1 KB (186 words) - 17:00, 5 February 2019
  • Arithmetic
    ...[subtraction]], [[multiplication]] and [[division]] with smaller values of numbers.
    561 bytes (76 words) - 12:53, 18 January 2019
  • Addition
    In [[mathematics]]: to sum 2 numbers. It is normally symbolized by the plus sign '+'.
    333 bytes (43 words) - 16:55, 29 August 2022
  • Multiplication
    ...sult of a multiplication is called the product of a and b, and each of the numbers is called a [[factor]] of the product ab. The result of multiplying no numbers (empty product) is always 1 (the multiplicative identity, see below). The m
    2 KB (271 words) - 17:00, 29 August 2022
  • Factorial number
    ...r a number, it represents multiplying a number by all [[whole number|whole numbers]] smaller than it.
    729 bytes (93 words) - 13:40, 5 November 2023
  • Factor
    A '''factor''' is one of the numbers or expressions that make up another number by [[multiplication]]. Let a and
    576 bytes (107 words) - 19:03, 5 February 2019
  • Subtraction
    ...n for finding the difference between two numbers. The special names of the numbers in a subtraction expression are, minuend &minus; subtrahend = difference. T
    893 bytes (128 words) - 16:58, 29 August 2022
  • Division
    ...numerator'' and ''denominator''). A fraction is an accepted way of writing numbers. It is not always expected that the result of the division is written in de
    2 KB (368 words) - 16:58, 29 August 2022
  • Factoring Database
    The '''Factoring Database''' is a database of [[factor]]s of numbers of any kind, programmed by Markus Tervooren. *Users can search for known factors of numbers
    1 KB (144 words) - 13:44, 24 January 2019
  • Factorization
    ...]]: Asymptotically faster than trial factoring, but the overhead for small numbers makes this method convenient only for finding factors in the range of 10 to ...ction factorization algorithm]] or CFRAC: It is a fast method to factorize numbers in the range 10 to 20 digits.
    4 KB (642 words) - 12:57, 5 March 2019
  • EFF prizes
    ...tion]] (EFF) offers prizes to the people/projects that finds the following numbers:
    2 KB (321 words) - 18:50, 14 December 2023
  • Mlucas
    ...mersenneforum.org/showthread.php?t=18748 Use of Mlucas code to test Fermat numbers] at [[MersenneForum]]
    1 KB (198 words) - 07:28, 22 August 2019
  • Prime95
    ...istributed computing]] project dedicated to finding new [[Mersenne prime]] numbers. More specifically, Prime95 refers to the Windows and Mac OS X versions of ...ne of the earliest [[grid computing]] projects, researching Mersenne prime numbers, to demonstrate distributed computing software of Entropia, a company he fo
    11 KB (1,586 words) - 12:24, 7 August 2021
  • Even number
    All numbers ending in 0, 2, 4, 6, or 8 are even.
    425 bytes (61 words) - 11:19, 7 March 2019
  • Odd number
    ...sed in decimal notation, the odd numbers end in 1, 3, 5, 7 or 9. All prime numbers except 2 are odd.
    316 bytes (42 words) - 11:21, 7 March 2019
  • CUDALucas
    2 KB (275 words) - 11:11, 21 August 2019
  • Glucas
    3 KB (426 words) - 14:21, 14 February 2019
  • CPU
    *[[Addition|Add]] two numbers together ...OS. A computer program can control these peripherals by reading or writing numbers to special places in the computer's memory.
    2 KB (366 words) - 09:57, 13 February 2019
  • Fast Fourier transform
    17 KB (2,684 words) - 18:50, 28 September 2023
  • ClLucas
    1 KB (137 words) - 18:48, 14 December 2023
  • GpuLucas
    ...rete weighted transform|IBDWT]]-method for fast multiplies modulo Mersenne numbers.
    2 KB (239 words) - 11:12, 13 February 2019
  • GpuOwL
    1 KB (216 words) - 05:22, 1 December 2020
  • Classes of computers
    ...ating point operation is the calculation of mathematical equations in real numbers. In terms of computational capability, memory size and speed, I/O technolog
    4 KB (558 words) - 22:55, 3 February 2019
  • MersenneForum
    2 KB (293 words) - 17:33, 5 July 2019
  • Primo
    '''Primo''' is a computer program which tests numbers for [[prime|primality]] using the [[Elliptic Curve Primality Proving]] (ECP
    1 KB (191 words) - 20:33, 12 May 2020
  • M33
    *[https://www.mersenne.org/primes/ List of known Mersenne prime numbers] at [[PrimeNet]]
    814 bytes (97 words) - 08:38, 18 February 2019
  • 2 Plus Tables
    ==Factorizations Of Cunningham Numbers C<sup>+</sup>(2,n) = 2<sup>n</sup> + 1==
    2 KB (127 words) - 15:28, 17 August 2019
  • Real number
    ...either a [[rational number]] or an [[irrational number]]. The set of real numbers is denoted by <math>\mathbb{R}</math>. ...math> can be constructed from <math>\mathbb{Q}</math>, the set of rational numbers using Dedekind cuts.
    390 bytes (57 words) - 15:00, 26 March 2023
  • Rational number
    ...denominator''') is an integer different from zero. The set of all rational numbers is named <math>\mathbb{Q}</math>. ...h>a/b</math> is called '''fraction'''. A fraction is irreducible when both numbers are [[coprime]], otherwise it can be reduced to an irreducible form by divi
    3 KB (541 words) - 15:01, 26 March 2023
  • Irrational number
    ...o mathematician takes that to be a definition. Some examples of irrational numbers are <math>\sqrt{2}</math> or <math>e</math>.
    763 bytes (124 words) - 15:14, 26 March 2023
  • Binary
    ...ernary, quaternary, and so on. Binary numeral system, a representation for numbers using only two [[digit]]s (usually, 0 and 1). Thus it is a [[base]] 2 numbe ...This makes them [[repunit]] numbers. This innate 'binariness' of Mersenne numbers makes calculations in the search for [[Mersenne prime]]s a bit easier.
    1 KB (210 words) - 11:16, 22 January 2019
  • Repunit
    ...t ('''rep'''eated '''unit''', "1" being the number referred to as "unity") numbers. 111 is a repunit, in base 2 it is equal to 7 (base 10), in base 3 it is eq Repunit numbers are of the form:
    1 KB (207 words) - 08:04, 12 March 2024
  • Parallel computing
    ...ld all be done in parallel. This would cut a 5 step procedure to 3. If the numbers were each 100 digits long and 10 individuals (or cores in a computer) worke
    3 KB (416 words) - 06:47, 1 May 2019
  • Long multiplication
    If a positional numeral system is used, a natural way of multiplying numbers is taught in schools as '''long multiplication''', sometimes called '''grad ...in base 2. [[Prime95]] does not use this form of multiplication for large numbers, using [[Fast Fourier transform|FFT]]'s is much faster. A person doing long
    2 KB (165 words) - 17:01, 29 August 2022
  • Trial factoring
    The simplest approach is to already have available a supply of small prime numbers to use as trial divisors. If P(i) is the i'th prime number so P(1) = 2, P(2 ...e [[Sieve of Eratosthenes]], itself requiring a small table of known prime numbers to start its process, such as 2 and 3.
    7 KB (1,221 words) - 13:20, 11 February 2019
  • Natural number
    ...er to physical objects. A farmer counting his sheep would only use natural numbers.
    316 bytes (43 words) - 15:00, 26 March 2023
  • Richard Crandall
    ...ade available for purchase posters of the largest known [[Mersenne prime]] numbers. Posters of [[M38]], [[M39]], [[M40]], [[M41]], [[M42]], [[M43]], [[M44]], *with [[Carl Pomerance]]: ''Prime numbers: A Computational Perspective.'' Springer 2001.
    3 KB (431 words) - 11:36, 14 January 2024
  • Irrational base discrete weighted transform
    ...f ''Rapid multiplication modulo the sum and difference of highly composite numbers.''] Math. Comp. 72:387-395, 2003. ...[http://thales.doa.fmph.uniba.sk/macaj/skola/teoriapoli/primes.pdf ''Prime numbers: A Computational Perspective: 2nd edition'']. Springer, 2005.
    1 KB (172 words) - 18:49, 28 September 2023
  • PrimeKit
    ...matica implementations of all 112 algorithms discussed in the book ''Prime Numbers: A Computational Perspective'' (2001) by [[Richard Crandall]] and Carl Pome
    1 KB (125 words) - 09:38, 23 January 2019
  • Coprime
    ...is 1 (<math>\gcd{(x,y)} = 1</math>). This does not mean that any of these numbers is prime. :Two random numbers are coprime with a probability over 60% (the exact number is <math>6/\pi^2<
    738 bytes (112 words) - 09:50, 23 January 2019
  • Greatest common divisor
    The '''Greatest common divisor (gcd)''' of two numbers, commonly expressed by <math>gcd(a, b)</math>, where <math>a</math> and <ma ...e [[coprime]] or relatively prime. This does not mean that either of these numbers are prime.
    2 KB (339 words) - 18:38, 27 September 2023
  • Modular arithmetic
    ...s;4 = 8 (because 5+5+5+5 = 8). This is arithmetic modulo 12 and the set of numbers representing the hours 0, 1, 2, 3,..., 11 is known as <b>Z</b>/12<b>Z</b>. ...We use the congruence symbol (<math>\equiv</math>) instead. Note that two numbers ''A'' and ''B'' are said to be congruent modulo ''n'' if ''A''-''B'' is a m
    4 KB (625 words) - 10:25, 23 January 2019
  • Elliptic curve method
    ...icance of this to the Elliptic Curve Method is that a huge amount of other numbers will have factors in common with our highly composite number. ...bound]] B<sub>1</sub>, we multiply the original point '''P''' by all prime numbers less than B<sub>1</sub> (each prime number is raised to a power such that t
    19 KB (3,181 words) - 22:27, 6 July 2023
  • University of California, Los Angeles
    ...090 [[Classes of computers#Mainframe computers|mainframe]]. Each of these numbers had over 1200 digits.
    2 KB (347 words) - 14:54, 19 September 2021
  • Raphael M. Robinson
    4 KB (526 words) - 14:51, 19 September 2021
  • Gordon Spence
    ...'', my family (not in that order!!) and of course, searching for big prime numbers."
    686 bytes (96 words) - 00:10, 15 January 2024
  • Wacław Sierpiński
    592 bytes (86 words) - 00:38, 15 January 2024
  • Sierpiński number
    These numbers are named after [[Wacław Sierpiński]].
    324 bytes (48 words) - 13:37, 8 April 2023
  • Sierpiński problem
    Consider numbers of the form {{V|N}} = {{Kbn|+|k|n}}, where {{Vk}} is odd and {{Vn}} > 0. If ...ence {{Kbn|+|78557|n}} can be prime. The same arguments can be said of the numbers 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965
    5 KB (650 words) - 10:25, 26 March 2024
  • Covering set
    ...'''covering set''' for a sequence of integers refers to a set of [[prime]] numbers such that every term in the sequence is divisible by at least one member of *[[Riesel_2_Riesel|Riesel numbers]]
    380 bytes (56 words) - 10:27, 26 March 2024
  • Seventeen or Bust
    ...P takes so much computational power, we try to eliminate as many non-prime numbers as possible from the queue by [[sieving]], which means to take a (relativel As of April 2010, Seventeen Or Bust has discovered eleven huge prime numbers. The four largest discoveries ranks as the tenth to thirteenth largest prim
    3 KB (544 words) - 16:44, 21 July 2019
  • Perfect number
    ...= 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128. These first four perfect numbers were the only ones known to the ancient Greeks.
    6 KB (885 words) - 11:33, 7 March 2019
  • Residue
    ...dues (meaning they both missed a prime) out of a pool of ~ 18.4 pentillion numbers, this is considered to be impossible.
    1 KB (235 words) - 10:24, 6 February 2019
  • GIMPS clients
    ...mbers in a time that any human being would want to wait for. That is, once numbers get really big, it is worthwhile to set up the multiplication using FFTs an ...S]], the [[distributed computing]] project researching [[Mersenne prime]]s numbers, uses. It is entirely [[Wikipedia:Computer console|console]]-based, with no
    8 KB (1,218 words) - 15:37, 13 August 2020
  • Mfaktc
    ...1 was released, this version added support for trial factoring on Wagstaff numbers.
    5 KB (765 words) - 14:54, 25 February 2019
  • Riesel number
    *{{Num|15000}} Riesel numbers in the {{OEIS|l|A101036}} *[[Riesel 2 Riesel|Riesel numbers]]
    827 bytes (112 words) - 08:21, 25 March 2024
  • M12
    ...[[bit level]] over 169.4. The current version of [[Prime95]] cannot handle numbers this large, nor can [[mfaktc]].
    2 KB (354 words) - 14:52, 19 September 2021
  • Primality test
    ==Primality tests for numbers {{V|N}} with special form== *[[Pépin's test]]: Used to test primality in Fermat numbers.
    3 KB (501 words) - 05:20, 3 August 2021
  • Square root
    ...the principle square root and the negative square root. For negative real numbers, the concept of [[imaginary number|imaginary]] and [[complex number]]s has ...roots of positive [[integer]]s are often ''[[irrational number]]s'', i.e., numbers not expressible as a [[quotient]] of two integers. For example, <math>\sqrt
    13 KB (1,873 words) - 16:52, 24 October 2020
  • Primality testing program
    ! Program !! Numbers tested !! Hardware !! OS !! Link | k &times; b<sup>n</sup>±c general numbers
    2 KB (314 words) - 21:23, 29 August 2019
  • Pépin's test
    ...f [[Fermat number]]s, but it is of no help for finding the factors of such numbers. Pépin's test can also be used for proving the primality of other numbers, like the [[Generalized Fermat number]]s <math>F_{n,2} = 4^{3^n}+2^{3^n}+1<
    2 KB (401 words) - 14:40, 6 March 2019
  • Proof by induction
    ...mathematical theorems that takes advantage of the structure of the natural numbers as described by the [[Peano postulates]]. Proof by induction is normally pe
    4 KB (679 words) - 13:57, 20 February 2019
  • Mprime (Cray)
    ...used [[Fast Fourier transform]]s for the [[multiplication]] of very large numbers. This represented an advance over the software used by [[Landon Curt Noll]]
    639 bytes (92 words) - 12:02, 7 February 2019
  • Probable prime
    ...eger]] that satisfies a specific condition also satisfied by all [[prime]] numbers.}} ...eger]] that satisfies a specific condition also satisfied by all [[prime]] numbers. Different types of probable primes have different specific conditions. Whi
    2 KB (232 words) - 07:28, 12 March 2024
  • Pseudoprime
    ...acticably slow; however probabilistic primality tests can rapidly generate numbers which are "[[Probable prime|probably prime]]". The term "probably" is not t
    1 KB (155 words) - 20:32, 25 July 2020
  • Pentium
    ...into and out of the [[CPU]] is a major consideration for processing large numbers.
    2 KB (285 words) - 00:50, 30 January 2019
  • Floating-point unit
    ...ally designed to carry out operations on [[floating-point|floating point]] numbers. Typical operations are [[addition]], [[subtraction]], [[multiplication]],
    2 KB (323 words) - 06:49, 1 May 2019
  • Floating-point
    ...m for representing [[real number]]s which supports a wide range of values. Numbers are in general represented approximately to a fixed number of [[Significan ...tion of the radix point), so when stored in the same space, floating-point numbers achieve their greater range at the expense of precision.
    2 KB (294 words) - 22:56, 3 February 2019
  • Factor5
    ...ing]] of [[Mersenne number]]s. It is capable of trial factoring very large numbers, many billions of digits. ...mbers with a hexillion digits or larger. Because it can work on such large numbers, it is used for [[Operation Billion Digits]].
    1 KB (201 words) - 21:16, 25 January 2019
  • Complex number
    When the second element equals zero the complex numbers behaves as real numbers. That's why the first element of the complex number is known as the ''real ...on and the definitions above we can deduce all basic operations on complex numbers:
    2 KB (280 words) - 14:59, 26 March 2023
  • Luigi Morelli
    *[http://www.doublemersennes.org/ Double Mersenne numbers]
    1 KB (154 words) - 01:15, 15 January 2024
  • Operation Billion Digits
    ...m.org] boards, has written a trial factoring program that can handle these numbers. It's a sub-project of [[Lone Mersenne Hunters]].
    6 KB (918 words) - 16:28, 24 July 2020
  • GIMPS factoring and sieving
    ...sieve. For example, if the sieving process eliminates 95% of the composite numbers, it may make more sense to test the remaining 5% along with any prime poten ===Constraints on prime factors of Mersenne numbers===
    6 KB (962 words) - 10:08, 7 March 2019
  • Non-negative
    ...n-negative reals are all the [[real number]]s from zero upwards. All whole numbers are non-negative.
    421 bytes (66 words) - 22:51, 26 January 2019
  • Ten million digits
    ...[[decimal]] [[digit]]s) in June of 1999, the next [[EFF prizes]] for prime numbers was '''ten million decimal digits'''. [[Prime95]] had an optional [[worktype]] to test numbers that were at least {{Num|10000000}} digits added (exponents of {{Num|332192
    979 bytes (146 words) - 14:23, 6 March 2019
  • LLR
    ...Primality testing program|program]] available to perform primality test on numbers of the form {{Vk}}•2<sup>{{Vn}}</sup>±{{V|c}}. *the fastest algorithms are for base two numbers (with {{Vk}} < 2<sup>{{Vn}}</sup>):
    2 KB (300 words) - 22:00, 16 December 2023
  • Mmff
    355 bytes (45 words) - 00:00, 27 January 2019
  • Double Mersenne number
    ...seeable future. Efforts are under way attempting to find factors for these numbers. Currently MM(61) and MM(127) are getting the most attention. ...e Mersennes. [[Luigi Morelli]] is co-ordinating the effort to factor these numbers.
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  • P-1 factorization method
    ...so that <math>2^{E_2}</math> is about B1 and the same for the other prime numbers. ''B'' is the greatest prime number less than or equal to B1. Most of the composite numbers have a prime factor much greater than the other prime factors. Suppose that
    5 KB (814 words) - 01:35, 12 March 2019
  • Self-initializing quadratic sieve
    ...ath>t^2 \equiv u\,\pmod N</math> where u is the product of small [[prime]] numbers. The set of these primes is the ''factor base''. These relations will be fo ...ecause the logarithms are rounded and we do not sieve with powers of prime numbers), perform a trial division of <math>g_{a,b}(x)/a</math> by the elements of
    10 KB (1,763 words) - 02:56, 12 March 2019
  • Special number field sieve
    ...T]] (a volunteer [[distributed computing]] effort) and others to factorise numbers of the [[Cunningham project]].
    1 KB (186 words) - 12:07, 19 February 2019
  • Lone Mersenne Hunters
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  • Fermat pseudoprimality test
    ...mbers, known as Carmichael numbers, for which the result is 1 whenever the numbers ''a'' and ''N'' are [[coprime]]. The smallest Carmichael number is 561. ...utational complexity, and which does not have the equivalent of Carmichael numbers.
    1 KB (164 words) - 10:56, 6 February 2019
  • Miller-Rabin pseudoprimality test
    The '''Miller-Rabin pseudoprimality test''' is based in two facts for prime numbers:
    3 KB (432 words) - 15:33, 28 January 2019
  • Generalized Fermat number
    John Cosgrave has studied the following numbers: Numbers of the form: <math>F_{n,r} = \sum_{i=0}^{p-1} \ 2^{i p^{n}} \ = \ 2^{(p-1)p
    5 KB (726 words) - 09:57, 12 September 2021
  • PrimeGrid
    ...Grid''' is a [[distributed computing]] project for searching for [[prime]] numbers of world-record size. It makes use of the [[BOINC|Berkeley Open Infrastruct
    3 KB (458 words) - 10:28, 26 March 2024
  • Worktype
    ...his is used to find factors. Currently this is only being assigned for low numbers that are known to be composite. *World record sized numbers to PRP test: Similar to World Record sized number to LL test, but uses the
    4 KB (603 words) - 02:31, 18 August 2019
  • 100 Million Digit Prefactor Project
    Uncwilly has resumed a leadership role in prefactoring numbers in this range. Many other users are now signed up to [[Lucas-Lehmer test|LL
    3 KB (534 words) - 10:06, 7 March 2019
  • FermatSearch
    361 bytes (46 words) - 13:48, 29 January 2019
  • Sieve of Eratosthenes
    The '''sieve of Eratosthenes''' is a method to find all [[prime]] numbers smaller than a given integer <math>N</math>. It's invention is credited to ...be divisible by <math>2</math>, we cross out every second number; all such numbers are composite.
    4 KB (654 words) - 11:10, 6 February 2019
  • Aliquot sequence
    An '''aliquot sequence''' is a sequence of numbers generated from an initial number using the sigma <math>\sigma(n)</math> fun The naive way to find the divisors of a number are to check the numbers from 1 to <math>\sqrt{n}</math> to see if they divide the number. Easy to d
    6 KB (914 words) - 19:49, 21 February 2023
  • Abundant number
    ==Abundant numbers and aliquot sequences== Abundant numbers increase the size of an [[aliquot sequence]] because when an abundant numbe
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  • Prime
    ...greater than 1 that is only divisible by itself and 1. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19. ...er Q, larger than P, that is equal to the product of the consecutive whole numbers from 2 to P plus the number 1. In other words, Q = (2 x 3 x 4 x 5 ... x P)
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  • Nomenclature and notation
    Either [[Marin Mersenne]] or one of the special class of numbers that bear his name. ...omposite]] or [[prime]] of the form <math>2^{x}-1</math>. For one of these numbers to be prime, <math>x</math> (the exponent) must also be prime. Thus, the no
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  • Seventeen or Bust/Primes
    These are the [[prime|prime numbers]] found by the [[Seventeen or Bust]] project so far.
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  • Seventeen or Bust/Factoring
    The programs commonly used to P-1 factor these numbers are [[Prime95]] and [http://linux.redbird.com/~alien88/sbfactor12.zip SBFac
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  • Why participate in GIMPS?
    ...28=1+2+4+7+14). He realized that the even perfect numbers (no odd perfect numbers are known) are all closely related to the primes of the form 2<sup>p</sup>- ...elementary number theory was developed while deciding how to handle large numbers, how to characterize their [[factor]]s and discover those which are prime.
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  • Sieving
    '''Sieving''' is an algorithm to discover [[smooth number]]s and [[prime]] numbers from a sequence of [[integer]]s much faster than [[trial factoring]], even The next step depends on whether we need to find prime number or smooth numbers.
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  • Sieving program
    *[[Gcwsieve]] (performing sieving of generalized Cullen/Woodall numbers n &times; b<sup>n</sup>+-1) http://sites.google.com/site/geoffreywalterreyn *[[PPSieve]] (sieving for factors of numbers of the form K &times; 2<sup>n</sup> + 1 or - 1. Independent of K's, but goo
    2 KB (220 words) - 11:42, 7 March 2019
  • NewPGen
    ...by Paul Jobling that is used to [[Sieving|sieve]] a set of many candidate numbers, of the following forms: ...t any k or n divisible by small primes. Since it works with a large set of numbers and uses fast implementations, it is a lot better than performing [[trial f
    3 KB (529 words) - 09:32, 7 March 2019
  • Proth prime
    A '''Proth prime''' is not a true class of numbers, but primes in the form {{Kbn|+|k|n}} with 2<sup>''n''</sup> > ''k'' are of
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  • Riesel Sieve
    ...nd primes for all the remaining k values to prove that they are not Riesel numbers.
    2 KB (326 words) - 10:29, 26 March 2024
  • Introductory stuff
    *[[Elliptic curve method|ECM]] curves on Mersenne numbers and Fermat numbers ...basically the same as a standard primality test, but test the primality of numbers with more than 100 000 000 digits. These work units take a very long time t
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  • What use are prime numbers anyway?
    ==So it would seem to be a fair question "What use are prime numbers?"== ...e would not exist as we know it, and all of this security depends on prime numbers.
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  • Glossary
    *'''[[Fermat number]]''' - Numbers of the form <math>2^{2^n} + 1</math>. *'''[[Generalized Fermat number]]''' - There are different kinds of such numbers.
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  • Overclocking
    ...mark, which uses 19 iterations in the test, is set 1 Million digits. Lower numbers denote faster calculation times (seconds), and hence, better performance.
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  • Triangular number
    ...angular) after a number) represents the summing of a number with all whole numbers smaller than it.
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  • Pocklington's theorem
    ...mada. It includes a program written in C language that proves primality of numbers of several thousand digits using this theorem.
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  • Multiple polynomial quadratic sieve
    ...v u\,\pmod N</math> where <math>u</math> is the product of small [[prime]] numbers. The set of these primes is the ''factor base''. These relations will be fo ...ecause the logarithms are rounded and we do not sieve with powers of prime numbers), perform a trial division of <math>g_{a,b}(x)/a</math> by the elements of
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  • Proth number
    ...<sup>{{Vn}}</sup> > {{Vk}}, all odd integers greater than 1 would be Proth numbers, but most pages lists them, too. ...nd [[Fermat number]]s ({{Kbn|+|2<sup>n</sup>}}) are special forms of Proth numbers.
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  • 27121 Search
    ...meGrid]], is a [[Distributed computing]] project, which searches for prime numbers of the form: 27 &times; 2<sup>n</sup> ± 1 and 121 &times; 2<sup>n</sup> ±
    983 bytes (138 words) - 13:25, 8 February 2019
  • GIMPS chances of finding a prime number
    How does this apply to finding [[prime]] numbers? ...there isn't a single prime between 31,397 and 31,469, a gap of seventy-one numbers. But this thinning out does not happen evenly like a river broadening as it
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  • Bit level
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  • Strong law of small numbers
    ...athematician]] Richard K. Guy published a paper ''"The Strong Law of Small Numbers"''. In it he states, :"There aren't enough small numbers to meet the many demands made of them."
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  • P+1 factorization method
    ...ent the value <math>M</math> in binary form. Then erase the first "1" (all numbers when represented in binary start with the digit "1"). Then for every "1" wr
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  • General number field sieve
    ...most efficient classical [[Factorization|factoring method]] for 100+ digit numbers.}}
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  • SNFS polynomial selection
    ==Cyclotomic numbers== For cyclotomic numbers, there are four standard operations you can perform on the number to derive
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  • NFSNET
    ...er field sieve|SNFS]] [[factorization]] methods to completely factor large numbers of interest to the math community. This project is now dead and replaced by ...zation]]s completed by NFSNET, all of them [[Cunningham project|Cunningham numbers]], are summarized below.
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  • Mersenneplustwo factorizations
    ...a prime, as well as simultaneously 2<sup>p</sup>-1 being prime). All such numbers are divisible by 3 since 2<sup>p</sup>-1 is not divisible by 3 (it's assume Participants use [[ECMclient]] to automatically download numbers, do ECM curves on them and upload them again.
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  • ECMclient
    ...al projects make use of ECMclient in their factoring of different kinds of numbers, including:
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  • GMP-ECM
    '''GMP-ECM''' is a program capable of factoring numbers using the [[Elliptic curve method|ECM]], [[P-1 factorization method|p-1]] a
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  • Gerbicz error checking
    ...um.org/showthread.php?t=22510 Original proposal of the technique for Proth numbers] ...hread.php?t=22471&p=465431 Original proposal of the technique for Mersenne numbers]
    3 KB (528 words) - 14:59, 3 October 2023
  • MPZ APR-CL
    ...ed on systems that support 64-bit data type. It can prove the primality of numbers up to 6021 digits long. ...systems that only support 32-bit data type. It can prove the primality of numbers up to 3827 digits long.
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  • ECPP-DJ
    ...plementation of ECPP test. However, Primo runs much faster for 1000+ digit numbers, especially on multi-core machines (ECPP-DJ is single threaded).
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  • GIMPS PrimeNet reports
    Details known factors of Mersenne numbers However, if a Mersenne number has been fully factored the largest factor is Displays the results of the PRP tests on cofactors of Mersenne numbers. Information shown here is similar to that shown on the PRP Results report,
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  • 15k Search
    ...o the rate at which Proth/PRP/PFGW can perform a [[primality test]] on the numbers. Actually, you should stop a little sooner - the idea is to find primes, af
    3 KB (517 words) - 14:51, 15 February 2019
  • On-Line Encyclopedia of Integer Sequences
    ...equence, the Encyclopedia search engine shows all sequences that match the numbers entered by the user. It can also find sequences given some text.
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  • Milestones report
    *[[Double check]]ing all numbers less than a given Mersenne Prime, thus proving its place in the sequence of *Testing all numbers less than a given Mersenne Prime once.
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  • Mersenne composite
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  • PrimeNet summary report
    ...The first row is for the numbers 0 - {{Num|999999}}. The second row is for numbers {{Num|1000000}} - {{Num|1999999}}. Etc. ...his is ''not'' the number of [[Mersenne prime]]s in the range. This is the numbers that need to be tested; all others need not be tested, because it is imposs
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  • Square number
    ..., 9 is a square number since it can be written as 3 &times; 3. If rational numbers are included, then the ratio of two square integers is also a square number ...two consecutive [[triangular number]]s. The sum of two consecutive square numbers is a centered square number. Every odd square is also a centered octagonal
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  • Editing
    <nowiki>"The [[Sierpiński problem]] is about numbers of the form <math>k*2^n+1</math>"</nowiki> : "The [[Sierpiński problem]] is about numbers of the form <math>k*2^n+1</math>"
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  • GIMPS milestones
    | 1998-12-26 || All Mersenne numbers less than {{Num|1000000}} digits tested at least once.
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  • Tutorial LLR
    [[LLR]] is a program used to prove primality of numbers. It can be rather slow (but faster if you support [[SSE2]]), and that's why ...'''you operate PRP that exact same way as LLR''' except that it only finds numbers that are [[probable prime]]s.
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  • Peano postulates
    #Let {{V|M}} be a set of natural numbers with the following properties: Then {{V|M}} contains all natural numbers.
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  • LL test distribution
    ...s tests are completed per unit time. There is an effort for people to test numbers in the 100 million digit range, which is the next digit size that is eligib
    2 KB (415 words) - 21:32, 12 February 2020
  • Mersenne.ca
    ...e probability of finding a factor is very low. By doing P-1 tests on these numbers, it was hoped to save unnecessary double-checks. At that time (before Prime These reports can be easily accessed with the abbreviated URL ''mersenne.ca/<numbers>'' where if the value is less than 2^32 it is assumed to be the exponent, o
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  • Value k
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  • Aurifeuillian factor
    *Numbers of the form <math>2^{4k+2}+1</math> have the following '''Aurifeuillian fac *Numbers of the form <math>b^n - 1</math> or <math>\Phi_n(b)</math>, where <math>b =
    10 KB (1,257 words) - 08:04, 24 June 2019
  • Factoring program
    *[[Factor5]] (performing Trial factoring of Mersenne numbers) http://www.moregimps.it/billion/download1.php
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  • Mtsieve
    *fbncsieve: search for factors of numbers in the form {{Kbn|+|k|b|n}} and {{Kbn|k|b|n}} for fixed b and n and variabl *kbbsieve: search for factors of numbers of the form {{Kbn|+|k|b|b}} or {{Kbn|k|b|b}} for fixed k and variable b
    2 KB (338 words) - 06:58, 28 March 2023
  • Available Double-Triple checks
    703 bytes (67 words) - 21:09, 16 March 2019
  • Carol-Kynea prime
    '''Carol numbers''' and '''Kynea numbers''' are numbers of the form <math>(b^n-1)^2-2</math> and <math>(b^n+1)^2-2</math>, respecti ...he form of these numbers, they are also classified as near-square numbers (numbers of the form <math>n^2-k</math>).
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  • Riesel 2 Low-weight
    {{DISPLAYTITLE:Riesel numbers of the form {{Kbn|k|n}} with Nash weight < 1000}} Riesel numbers {{Kbn|k|n}} where the [[Nash weight]] is smaller than 1000.
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  • Williams prime MM 10
    |WiRemarks=These primes are [[Near-repdigit]] numbers; see [https://stdkmd.net/nrr/8/89999.htm Factorizations of 899...99] by [[M
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  • Williams prime MM 14
    *John Eisenmann: [http://ostracodfiles.com/primes14/primes.php "PRIME NUMBERS OF THE FORM A*14^B-1"]
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  • Proth prime small bases least n
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  • GMP-Fermat
    433 bytes (60 words) - 07:02, 21 June 2019
  • Proth.exe
    ...([[Cullen prime|Generalized Cullen]]/[[Woodall prime|Generalized Woodall]] numbers)
    667 bytes (101 words) - 16:44, 31 August 2021
  • NFS@Home
    # Numbers are proposed by people in the [https://www.mersenneforum.org/showthread.php # Greg Childers (maintainer of the project) puts the numbers in the queue to be sieved.
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  • Leyland prime P table
    |resultsheader=<b>There are %PAGES% numbers</b>:\n
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  • Paul Leyland
    ...[[Cunningham number]]s, [[Cullen number]]s, [[Woodall number]]s, etc., and numbers of the form <math>x^y + y^x</math>, which are now called [[Leyland number]]
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  • Leyland prime P csv
    |resultsheader=There are %PAGES% numbers:\n
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  • Leyland number
    ...who first studied these numbers in 1994. The first few nontrivial Leyland numbers are given by OEIS sequence {{OEIS|A076980}}. The second kind of numbers are of the form <math>x^y-y^x</math>.
    8 KB (906 words) - 09:59, 5 January 2023
  • Saouter number
    A '''Saouter number''' is a type of [[Generalized Fermat number]]. Numbers of this type have the form ...t-like sequence", 2005]</ref> after [[Yannick Saouter]], who studied these numbers<ref>[https://hal.inria.fr/file/index/docid/73966/filename/RR-2728.pdf Y.Sao
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  • Mersenne@home
    |purpose=To test Mersenne numbers for factors and primality
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  • 3 Minus Tables
    ==Factorizations Of Cunningham Numbers C<sup>-</sup>(3,n) = 3<sup>n</sup> - 1==
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  • James Heinrich
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  • Generalized Fermat 262144
    {{Navbox Generalized Fermat numbers}}
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  • PrimeGrid Sophie Germain Search
    The most recent iteration of the project searched for prime numbers of the form {{Kbn|k|1290000}}, for ''k'' ≤ 10<sup>13</sup>. The first version of the project searched for prime numbers {{Kbn|k|n}} for 666,666 ≤ ''n'' ≤ 666,691 with varying ranges of ''k''
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  • Homogeneous Cunningham M 11 5 281
    [[Category:Homogeneous Cunningham numbers]]
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  • Riesel 2 Riesel
    {{DISPLAYTITLE:Riesel numbers of the form {{Kbn|k|n}}}} ...are odd numbers {{Vk}} for which {{Kbn|k|n}} is composite for all natural numbers {{Vn}}.
    808 bytes (110 words) - 21:12, 17 December 2023
  • Riesel 2 Count-100
    {{DISPLAYTITLE:Riesel numbers of the form {{Kbn|k|n}} with 100 and more primes}} Riesel numbers {{Kbn|k|n}} with 100 or more prime values {{Vn}}.
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  • Riesel 2 15k-value
    {{DISPLAYTITLE:Riesel numbers of the form {{Kbn|k|n}} with {{Vk}} mod 15 = 0}} Riesel numbers {{Kbn|k|n}} where {{Vk}}-value is a multiple of 15.
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  • Riesel 2 2145k-value
    {{DISPLAYTITLE:Riesel numbers of the form {{Kbn|k|n}} with {{Vk}} mod 2145 = 0}} Riesel numbers {{Kbn|k|n}} where {{Vk}}-value is a multiple of 2145.
    1 KB (156 words) - 07:01, 16 July 2021
  • Riesel 2 2805k-value
    {{DISPLAYTITLE:Riesel numbers of the form {{Kbn|k|n}} with {{Vk}} mod 2805 = 0}} Riesel numbers {{Kbn|k|n}} where {{Vk}}-value is a multiple of 2805.
    1 KB (156 words) - 07:08, 16 July 2021
  • Riesel 2 Count-0
    {{DISPLAYTITLE:Riesel numbers of the form {{Kbn|k|n}} with no prime value so far}} Riesel numbers {{Kbn|k|n}} where no prime values are known.
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  • Liskovets-Gallot conjectures
    [[Yves Gallot]] extended this for {{Kbn|k|n}} numbers and gave also the first solutions as: ...w.primepuzzles.net/problems/prob_036.htm Problem 36] "The Liskovets-Gallot numbers" from [https://www.primepuzzles.net/index.shtml PP&P connection] by [[Carlo
    2 KB (367 words) - 12:42, 9 May 2024
  • CRUS Liskovets-Gallot
    ...w.primepuzzles.net/problems/prob_036.htm Problem 36] "The Liskovets-Gallot numbers" from [https://www.primepuzzles.net/index.shtml PP&P connection] by [[Carlo
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  • Riesel 2 3k-value
    {{DISPLAYTITLE:Riesel numbers of the form {{Kbn|k|n}} with {{Vk}} mod 3 = 0}} Riesel numbers {{Kbn|k|n}} where {{Vk}}-value is a multiple of 3.
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  • Statistics
    | [[:Category:Leyland prime P|Leyland numbers {{V|x}}<sup>{{V|y}}</sup>+{{V|y}}<sup>{{V|x}}</sup>]] || number || style="t | [[:Category:Leyland prime M|Leyland numbers {{V|x}}<sup>{{V|y}}</sup>-{{V|y}}<sup>{{V|x}}</sup>]] || number || style="t
    11 KB (1,385 words) - 17:23, 5 April 2024
  • Riesel prime 10 9992
    |RRemarks=All numbers are [[Near Repdigit]] 999199...99.
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  • Proth 2 3k-value
    {{DISPLAYTITLE:Proth numbers of the form {{Kbn|+|k|n}} with {{Vk}} mod 3 = 0}} Proth numbers {{Kbn|+|k|n}} where {{Vk}}-value is a multiple of 3.
    1 KB (156 words) - 09:18, 23 July 2021
  • ToDo
    *Create own template to calculate indexes for long numbers, see [[Riesel k=3k value]] *create template for (General) Fermat Numbers
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  • Riesel problem 2
    4 KB (386 words) - 06:41, 29 March 2024
  • Proth prime 2 121
    1 KB (112 words) - 19:02, 9 April 2023
  • Proth 2 15k-value
    {{DISPLAYTITLE:Proth numbers of the form {{Kbn|+|k|n}} with {{Vk}} mod 15 = 0}} Proth numbers {{Kbn|+|k|n}} where {{Vk}}-value is a multiple of 15.
    1 KB (156 words) - 09:22, 23 July 2021
  • Proth 2 2145k-value
    {{DISPLAYTITLE:Proth numbers of the form {{Kbn|+|k|n}} with {{Vk}} mod 2145 = 0}} Proth numbers {{Kbn|+|k|n}} where {{Vk}}-value is a multiple of 2145.
    1 KB (156 words) - 09:36, 23 July 2021
  • Proth 2 2805k-value
    {{DISPLAYTITLE:Proth numbers of the form {{Kbn|+|k|n}} with {{Vk}} mod 2805 = 0}} Proth numbers {{Kbn|+|k|n}} where {{Vk}}-value is a multiple of 2805.
    1 KB (158 words) - 09:16, 22 March 2024
  • Proth 2 Sierpinski
    {{DISPLAYTITLE:Sierpiński numbers of the form {{Kbn|+|k|n}}}} ...re odd numbers {{Vk}} for which {{Kbn|+|k|n}} is composite for all natural numbers {{Vn}}.
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  • Proth 2 Count-0
    {{DISPLAYTITLE:Proth numbers of the form {{Kbn|+|k|n}} with no prime value so far}} Proth numbers {{Kbn|+|k|n}} where no prime values are known.
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  • Proth 2 Count-100
    {{DISPLAYTITLE:Proth numbers of the form {{Kbn|+|k|n}} with 100 and more primes}} Proth numbers {{Kbn|+|k|n}} with 100 or more prime values {{Vn}}.
    916 bytes (122 words) - 07:51, 26 July 2021
  • Generalized Fermat number 2 1
    {{DISPLAYTITLE:Fermat numbers}} |resultsheader=<b>There are %PAGES% Fermat numbers</b>:\n
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  • GF Divisors 2 1
    595 bytes (83 words) - 22:51, 10 September 2021
  • Proth 2 Low-weight
    {{DISPLAYTITLE:Proth numbers of the form {{Kbn|+|k|n}} with Nash weight < 1000}} Proth numbers {{Kbn|+|k|n}} where the [[Nash weight]] is smaller than 1000.
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  • Generalized Fermat number 3 1
    {{DISPLAYTITLE:Generalized Fermat numbers 3<sup>2<sup>n</sup></sup>+1 div 2}} |resultsheader=<b>There are %PAGES% General Fermat numbers</b>:\n
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  • GF Divisors 3 1
    617 bytes (84 words) - 22:54, 10 September 2021
  • Readme.txt
    prime95.exe The windows program to trial factor and primality test Mersenne numbers. mprime The Linux program to trial factor and primality test Mersenne numbers.
    3 KB (454 words) - 12:28, 7 August 2021
  • Riesel prime 10 9993
    .../prime/prime_difficulty.txt List of near-repdigit-related (probable) prime numbers Studio Kamada] (see wlabel=99929w)
    763 bytes (88 words) - 00:11, 30 April 2024
  • Riesel problem 3
    4 KB (337 words) - 18:52, 30 April 2024
  • Even Riesel Problem
    Except a few numbers of the odd {{Vk}}'s of the Riesel problem a prime was found (mostly a highe
    7 KB (718 words) - 10:32, 26 March 2024
  • Proth prime 5 4
    718 bytes (75 words) - 07:17, 14 February 2023
  • Riesel prime 3 2
    ...4747-X/S0025-5718-1972-0314747-X.pdf H.C.Williams, C.R.Zarnke: "Some prime numbers of the forms 2A3^n+1 and 2A3^n-1"] Math. Comp. 26 (1972), 995-998<br>The {{
    4 KB (449 words) - 06:55, 30 May 2023
  • Riesel prime 3 4
    ...4747-X/S0025-5718-1972-0314747-X.pdf H.C.Williams, C.R.Zarnke: "Some prime numbers of the forms 2A3^n+1 and 2A3^n-1"] Math. Comp. 26 (1972), 995-998<br>The {{
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  • PrimeGrid Generalized Fermat Prime Search
    ...earching for [[Generalized Fermat number#Dubner|Generalized Fermat]] prime numbers.
    2 KB (310 words) - 06:38, 29 December 2023
  • Riesel problem 4
    4 KB (336 words) - 18:50, 30 April 2024
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