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  • Cunningham project
    ...it is considered the oldest continuously ongoing activity in computational number theory. ...exponent. The second type is [[aurifeuillian factor]], in which the whole number can be split into two parts directly, for certain combination of values of
    7 KB (1,150 words) - 23:48, 19 April 2023
  • M25
    | number=448679166119...353511882751 M25 is 2<sup>{{Num|21701}}</sup>-1, a number of {{Num|6533}} [[digit]]s.
    2 KB (303 words) - 11:01, 26 February 2019
  • Lucas-Lehmer test
    ...cas-Lehmer test''' is a deterministic algorithm used to prove a [[Mersenne number]] either composite or prime. It is the last stage in the procedure employed ...<sup>p</sup>-1 would divide into another number, now called a Lucas-Lehmer number denoted S<sub>n</sub> where S<sub>0</sub>=4 and S<sub>n</sub> = (S<sub>n-1<
    20 KB (3,572 words) - 14:30, 17 February 2019
  • Édouard Lucas
    ...[[Mersenne prime]] for almost 75 years, and is still the highest [[prime]] number discovered without the aid of a computer.
    2 KB (296 words) - 01:09, 15 January 2024
  • Edson Smith
    .... In August 2008, one of these computers found a [[M47| World record prime number.]] ...a.edu) discovered a new prime [[M47]]. It remained the largest known prime number for almost four and a half years.
    4 KB (564 words) - 00:11, 15 January 2024
  • Digit
    ...g fingers) of the hands correspond to the 10 symbols of the common base 10 number system, i.e. the [[decimal]] (ancient Latin adjective ''dec.'' meaning ten) In a given number system, if the [[base]] is an integer, the number of digits required is always equal to the absolute value of the base.
    1 KB (171 words) - 10:17, 18 January 2019
  • Decimal
    ...base 2. The length of a number (how many [[digit]]s it takes to write the number) depends upon the base.
    1 KB (190 words) - 10:23, 18 January 2019
  • Base
    In [[Mathematics]], a '''base''' or '''radix''' is the number of different [[digit]]s that a system of counting uses to represent numbers Bases must be a [[whole number]] bigger than 0. If it was 0, then there would be no digits.
    2 KB (399 words) - 10:37, 18 January 2019
  • Whole number
    The term '''whole number''' does not have a consistent definition. Various authors use it in one of *the positive integers (1, 2, 3, ...) (often called [[natural number]]s)
    413 bytes (54 words) - 09:51, 8 February 2019
  • Integer
    ...r ''Zahlen'' (German for "numbers"). They are also known as the '''[[whole number]]s''', although that term is also used to refer only to the positive intege [[Category:Number systems]]
    3 KB (404 words) - 14:58, 26 March 2023
  • Mathematics
    '''Mathematics''' is the science of space, number and quantity. ...theorem: If you subtract an odd number from an even number you get an odd number.
    1 KB (186 words) - 17:00, 5 February 2019
  • Multiplication
    '''Multiplication''' is the process of calculating the result when a number a is [[Addition|added]] to itself b times. The result of a multiplication i ...es are in [[exponent|exponentiation]] (<math>a^0=1</math>) and [[factorial number]]s (0!=1).
    2 KB (271 words) - 17:00, 29 August 2022
  • Exponent
    ...wer a base number is raised to, the exponent is the superscript value of a number written as <math>a^p</math>. ...duct]] a &times; a &times; a &times; a is written as <math>a^4</math>, the number 4 is the index, or exponent.
    1 KB (273 words) - 16:56, 29 August 2022
  • Factorial number
    ...or bang) after a number, it represents multiplying a number by all [[whole number|whole numbers]] smaller than it. *[[Multifactorial number]]
    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 b be integers. Then a divides b (which may ...a number that has factors other than itself and 1 is called a [[composite number]].
    576 bytes (107 words) - 19:03, 5 February 2019
  • Composite number
    ...he integers <math>a</math> and <math>b</math> are both greater than 1, the number is composite. *[[Wikipedia:Composite number|Wikipedia]]
    358 bytes (56 words) - 23:30, 26 October 2020
  • Subtraction
    If the minuend is less than the subtrahend, the difference will be a negative number. For example, 17 &minus; 25 = ( &minus;8 ). We can say this as, "Seventeen
    893 bytes (128 words) - 16:58, 29 August 2022
  • Factoring Database
    **Near Cunningham number **Near-repdigit
    1 KB (144 words) - 13:44, 24 January 2019
  • Factorization
    ...ality tests|primality test]], we have to attempt the factorization of this number. The same can be said of the other factor ''b''. So it can be seen that we ..."special purpose" methods whose execution time depends on the size (i.e., number of digits), or on other particular properties of the factors.
    4 KB (642 words) - 12:57, 5 March 2019
  • EFF prizes
    ...substantial award for the person that discovers a ten million digit prime number. ...00}}''' to the first individual or group who discovers a [[Megaprime|prime number with at least '''{{Num|1000000}} decimal digits''']] (awarded 2000-04-06)
    2 KB (321 words) - 18:50, 14 December 2023
  • Number theory
    '''Number theory''' is a branch of pure [[mathematics]] devoted primarily to the stud
    202 bytes (29 words) - 12:55, 20 January 2019
  • Mlucas
    ...st]]s of prime-exponent [[Mersenne number]]s, and Pépin tests of [[Fermat number]]s. It is written by [[Ernst Mayer]] using C programming language and [[ARM
    1 KB (198 words) - 07:28, 22 August 2019
  • M48
    | number=581887266232...071724285951 To confirm that there were no errors in the [[hardware]] or [[software]], the number had to be independently verified by running tests on various machines with
    2 KB (235 words) - 11:49, 18 February 2019
  • M47
    | number=316470269330...166697152511 ...er possible usages refer to the [[Nomenclature and notation]] article. The number now refered to as M47 was actually the 45th Mersenne prime found. [[M45]],
    5 KB (694 words) - 13:17, 21 August 2019
  • Prime95
    ...Awards]). As such, a user who uses Prime95 to discover a qualifying prime number would not be able to claim the prize directly. A free software package woul ...page4.html Web Archive]). This is slightly shorter than a 24 million digit number. Newer versions of Prime95 (version 25, 26 and 27) can handle Mersenne numb
    11 KB (1,586 words) - 12:24, 7 August 2021
  • Even number
    ...ber of objects can be placed into exactly 2 groups that have the identical number of objects. *[[Odd number]]
    425 bytes (61 words) - 11:19, 7 March 2019
  • Odd number
    An '''odd number''' is any [[integer]] that is not divisible by 2. *[[Even number]]
    316 bytes (42 words) - 11:21, 7 March 2019
  • CUDALucas
    ...UDA]]-based program written by [[Shoichiro Yamada]] for testing [[Mersenne number]]s for primality with [[Lucas-Lehmer test]].
    2 KB (275 words) - 11:11, 21 August 2019
  • Glucas
    ...a program to perform the [[Lucas-Lehmer test]] for primality on [[Mersenne number]]s. It is related to [[GIMPS]] effort to search the largest primes ever fou
    3 KB (426 words) - 14:21, 14 February 2019
  • M41
    | number=299410429404...882733969407 ...senneforum.org/txt/41.txt {{Num|7816230}} decimal digits] long. This prime number was the seventh record prime found by the [[GIMPS]] project.
    1 KB (203 words) - 11:26, 18 February 2019
  • M43
    | number=315416475618...411652943871 The number is [http://www.mersenneforum.org/txt/43.txt {{Num|9152052}} decimal digits]
    1 KB (191 words) - 11:31, 18 February 2019
  • M46
    | number=169873516452...765562314751 The number is [http://www.mersenneforum.org/txt/46.txt {{Num|12837064}} decimal digits
    2 KB (248 words) - 11:45, 18 February 2019
  • Distributed computing
    There are a number of individuals or groups in the fields of science, mathematics, cryptograph ...ng comes into the picture. Using the [[unused computing power]] of a large number of computers some of these problems can actually be solved within a reasona
    4 KB (674 words) - 12:11, 19 February 2019
  • CPU
    *Test to see if one number is larger than another *Move a number from one place to another
    2 KB (366 words) - 09:57, 13 February 2019
  • M42
    | number=122164630061...280577077247 ...senneforum.org/txt/42.txt {{Num|7816230}} decimal digits] long. This prime number was the eighth record prime found by the [[GIMPS]] project.
    934 bytes (118 words) - 11:26, 18 February 2019
  • M1
    | number= 3
    193 bytes (19 words) - 13:43, 17 February 2019
  • Conjectures 'R Us
    ...project]] in search for lowest [[Sierpiński number|Sierpiński]]/[[Riesel number|Riesel]] values.}} ...mbination of algebraic and trivial factor(s), or make [[Generalized Fermat number]]'s.
    3 KB (503 words) - 02:20, 1 May 2024
  • Fast Fourier transform
    Let ''x''<sub>0</sub>, ...., ''x''<sub>''n''-1</sub> be [[complex number]]s. The DFT is defined by the formula ...nd conquer algorithm that recursively breaks down a DFT of any [[Composite number|composite]] size ''n'' = ''n''<sub>1</sub>''n''<sub>2</sub> into many small
    17 KB (2,684 words) - 18:50, 28 September 2023
  • ClLucas
    ...penCL-based program written by [[Shoichiro Yamada]] for testing [[Mersenne number]]s for primality with [[Lucas-Lehmer test]]. It is an OpenCL counterpart of
    1 KB (137 words) - 18:48, 14 December 2023
  • GpuLucas
    ...[[CUDA]]-based program written by [[Andrew Thall]] for testing [[Mersenne number]]s for primality.
    2 KB (239 words) - 11:12, 13 February 2019
  • GpuOwL
    ...s a [[OpenCL]]-based program written by Mihai Preda for testing [[Mersenne number]]s for primality.
    1 KB (216 words) - 05:22, 1 December 2020
  • Classes of computers
    ...The architecture of modern supercomputers tend to be built around a large number of "off the shelf" [[CPU]]'s or [[GPU]]'s, rather than those of the past th
    4 KB (558 words) - 22:55, 3 February 2019
  • MersenneForum
    ...upport [[GIMPS]], the broader community of [[Mersenne number]]s, [[prime]] number, and factoring projects. In addition to being the de facto help and support ==Prime number software discussion and development==
    2 KB (293 words) - 17:33, 5 July 2019
  • Primo
    The program does not require a number to be of any specific form. If a number is found to be prime, a [[primality certificate]] is produced, which can be ...index.php?id=1100000001443762221 40,000 digits]. The certification of this number was done by [[Paul Underwood]] with Primo 4.3.0. The certification process
    1 KB (191 words) - 20:33, 12 May 2020
  • M32
    | number=174135906820...328544677887 :2<sup>756 839</sup>-1, a number {{Num|227832}} [[decimal]] [[digit]] long was found to be [[prime]] on 1992
    2 KB (279 words) - 08:35, 18 February 2019
  • M33
    | number=129498125604...243500142591 '''M33''' refers to 33rd [[Mersenne prime]] number 2<sup>{{Num|859433}}</sup>-1.
    814 bytes (97 words) - 08:38, 18 February 2019
  • M34
    | number=412245773621...976089366527 .../sup>-1, which is a number {{Num|378632}} [[decimal]] [[digit]]s long. The number was found to be prime in 1996.
    3 KB (513 words) - 08:42, 18 February 2019
  • Real number
    A '''real number''' is either a [[rational number]] or an [[irrational number]]. The set of real numbers is denoted by <math>\mathbb{R}</math>. *[[Wikipedia:Real_number|Real number]]
    390 bytes (57 words) - 15:00, 26 March 2023
  • Absolute value
    The '''absolute value''' of a [[real number]] is defined as: The absolute value of a [[complex number]] z = x + iy is defined as:
    556 bytes (89 words) - 16:58, 29 August 2022
  • Rational number
    A '''rational number''' is a [[real number]] which can be written as <math>\frac{a}{b}</math> or <math>a/b</math> wher ...r [[greatest common divisor]]. This operation does not change the rational number represented by the fraction.
    3 KB (541 words) - 15:01, 26 March 2023
  • Irrational number
    ...an '''irrational number''' is any [[real number]] that is not a [[rational number]], i.e., one that cannot be written as a ratio of two integers, i.e., it is *[[Wikipedia:Irrational_number|Irrational number]]
    763 bytes (124 words) - 15:14, 26 March 2023
  • Binary
    [[Mersenne number]]s when written in binary are all 1's. This makes them [[repunit]] numbers.
    1 KB (210 words) - 11:16, 22 January 2019
  • Repunit
    ...[[Mersenne number]]s are repunit ('''rep'''eated '''unit''', "1" being the number referred to as "unity") numbers. 111 is a repunit, in base 2 it is equal to | align="center"|[[Mersenne number]]<br>(Base 2 repunit)
    1 KB (207 words) - 08:04, 12 March 2024
  • Parallel computing
    ...e is no need to exchange data on each step, it is feasible to process each number on a physically separate machine.
    3 KB (416 words) - 06:47, 1 May 2019
  • Processor
    ...in a single chip package. A '''many-core''' processor is one in which the number of cores is large enough that traditional multi-processor techniques are no
    2 KB (269 words) - 14:56, 22 January 2019
  • Trial factoring
    ...are found, the number in question is prime; otherwise, it is a [[composite number]]. ..., P(2) = 3, P(3) = 5, etc, then the last prime factor possibility for some number N would be P(m) where P(m + 1) squared exceeds N.
    7 KB (1,221 words) - 13:20, 11 February 2019
  • M40
    | number=125976895450...762855682047 ...senneforum.org/txt/40.txt {{Num|6320430}} decimal digits] long. This prime number was the sixth record prime found by the [[GIMPS]] project.
    1 KB (189 words) - 11:17, 18 February 2019
  • M39
    | number=924947738006...470256259071 ...[[Michael Cameron]], using [[Prime95]] written by [[George Woltman]]. The number is [http://www.mersenneforum.org/txt/39.txt {{Num|4053946}} decimal digits]
    868 bytes (109 words) - 11:14, 18 February 2019
  • M44
    | number=124575026015...154053967871 ...senneforum.org/txt/44.txt {{Num|9808358}} decimal digits] long. This prime number was the tenth record prime found by the [[GIMPS]] project.
    997 bytes (129 words) - 11:35, 18 February 2019
  • M45
    | number=202254406890...022308220927 In an interview Hans-Michael Elvenich, a German electrical engineer and prime number enthusiast, stated: "After four years of searching for a prime on [[GIMPS]]
    2 KB (251 words) - 11:40, 18 February 2019
  • Natural number
    *[[Whole number]] *[[Wikipedia:Natural_number|Natural number]]
    316 bytes (43 words) - 15:00, 26 March 2023
  • Richard Crandall
    ...uter]] scientist and physicist who has made contributions to computational number theory. He received a doctorate from [[Massachusetts Institute of Technolog His Erdös number is 2. He was one of the primary verifiers of [[M32]], [[M33]], and [[M34]].
    3 KB (431 words) - 11:36, 14 January 2024
  • M38
    | number=437075744127...142924193791 ...[[Nayan Hajratwala]], using [[Prime95]] written by [[George Woltman]]. The number is [http://www.mersenneforum.org/txt/38.txt {{Num|2098960}} decimal digits]
    1 KB (165 words) - 11:10, 18 February 2019
  • Nayan Hajratwala
    *[http://primes.utm.edu/notes/6972593 And the winning number is...]
    809 bytes (109 words) - 23:55, 14 January 2024
  • Irrational base discrete weighted transform
    ...T''') is a variant of the [[Fast Fourier transform]] using an [[Irrational number|irrational]] base. It was proposed by [[Richard Crandall]] and [[Barry Fagi The IBDWT is used to perform FFT multiplication modulo [[Mersenne number]] in such programs as [[Prime95]], [[CUDALucas]], [[Glucas]], [[gpuLucas]].
    1 KB (172 words) - 18:49, 28 September 2023
  • PrimeKit
    ...optimized), but there is also an "Extras" folder containing some efficient number-theoretical C sources.
    1 KB (125 words) - 09:38, 23 January 2019
  • Coprime
    :Two random numbers are coprime with a probability over 60% (the exact number is <math>6/\pi^2</math>).
    738 bytes (112 words) - 09:50, 23 January 2019
  • Greatest common divisor
    ...re <math>a</math> and <math>b</math> are positive integers, is the maximum number that divides both <math>a</math> and <math>b</math>. There are faster methods, especially when number of thousands or millions of digits are used, as in [[GIMPS]], but they are
    2 KB (339 words) - 18:38, 27 September 2023
  • Modular arithmetic
    We can visualize this arithmetic using a clock. Suppose that the number 12 in the clock is replaced by zero. Then when we have to add an hour, we g where B is the number such that <math>A * B = 1</math> (mod <math>n</math>).
    4 KB (625 words) - 10:25, 23 January 2019
  • Montgomery multiplication
    The Montgomery representation of a number <math>a</math> is the value ...presentation to normal, just perform a Montgomery multiplication using the number 1 as the second factor.
    4 KB (582 words) - 17:01, 29 August 2022
  • Elliptic curve method
    ...le of a point on a random elliptic curve [[modular arithmetic|modulo]] the number to be factored. It is currently the best [[algorithm]] known, among those w ...ber]]. This method cannot be used when it is not known in advance that the number is composite, so it cannot be used as a [[primality test]].
    19 KB (3,181 words) - 22:27, 6 July 2023
  • M35
    | number=814717564412...868451315711 ...er]]. The [[Lucas-Lehmer test]] took 88 hours to run. The primality of the number was confirmed by Slowinski. This showed the effectiveness of [[distributed
    2 KB (224 words) - 11:00, 18 February 2019
  • University of California, Los Angeles
    ...75 years (2 in the very first day of the run, no less). And he raised the number of digits of the largest known [[prime]] (in general) and Mersenne Prime fr
    2 KB (347 words) - 14:54, 19 September 2021
  • Raphael M. Robinson
    ...0) proved that an essentially undecidable theory need not have an infinite number of axioms by coming up with a counterexample: Robinson arithmetic ''Q''. '' ...ity of California, Los Angeles]]. In 1952, he showed that these [[Mersenne number]]s were all composite except for 17 values of ''n'' = 2, 3, 5, 7, 13, 17,
    4 KB (526 words) - 14:51, 19 September 2021
  • M36
    | number=623340076248...743729201151 ...k book. It took Spence's 100 MHz [[Pentium]] computer 15 days to prove the number prime. Alan White Managing Director at Technology Business Solutions, who p
    2 KB (279 words) - 11:01, 18 February 2019
  • Jonathan Pace
    ...umber-discovered How a FedEx employee discovered the world's largest prime number]. ''Popular Science''. 11 Jan 2018. ...-employee.html FedEx employee from Tennessee discovers largest known prime number]. ''CNBC''. 5 Jan 2018.
    2 KB (242 words) - 00:08, 15 January 2024
  • Wacław Sierpiński
    ...h|1882-03-14|1969-10-21}} was a Polish [[mathematician]] contributing in [[number theory]] and others. ...proved there are infinitely many odd integers {{Vk}} (named [[Sierpiński number]]s after him) such that {{Kbn|+|k|n}} is composite for all {{Vn}}.
    592 bytes (86 words) - 00:38, 15 January 2024
  • Sierpiński number
    ...l number {{Vk}} such that all {{Kbn|+|k|n}} for all {{Vn}} are [[Composite number|composite]]. *[[Wikipedia:Sierpiński_number|Sierpiński number]]
    324 bytes (48 words) - 13:37, 8 April 2023
  • Sierpiński problem
    The '''Sierpiński problem''' in [[number theory]] was proposed by [[Wacław Sierpiński]] in 1960. ...[[composite number]] {{V|N}}, then {{Vk}} is said to be a '''[[Sierpiński number]]'''.
    5 KB (650 words) - 10:25, 26 March 2024
  • Seventeen or Bust
    ...(SOB)''' was a [[distributed computing]] project working on a problem in [[number theory]] called the [[Sierpiński problem]]. It is currently a subproject o ...here ''k'' is one of the remaining 17 (now 5) candidates for [[Sierpiński number]]s smaller than 78557, and ''n'' a positive integer. In order to find such
    3 KB (544 words) - 16:44, 21 July 2019
  • M2
    | number= 7
    193 bytes (19 words) - 13:43, 17 February 2019
  • M3
    | number= 31
    194 bytes (19 words) - 13:43, 17 February 2019
  • M4
    | number= 127
    195 bytes (19 words) - 13:44, 17 February 2019
  • M5
    | number= 8191
    204 bytes (18 words) - 13:46, 17 February 2019
  • Perfect number
    In [[mathematics]], a '''perfect number''' is defined as an integer which is the sum of its proper positive divisor ...and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128.
    6 KB (885 words) - 11:33, 7 March 2019
  • M9
    | number=2305843009213693951 ...l factoring]]. Pervushin used the [[Lucas-Lehmer test]] to prove that this number is prime.
    2 KB (213 words) - 14:30, 17 February 2019
  • Double check
    *human error (entering wrong number to test, misreading data, etc.) ...t]] does a verfication on all [[factor]]s reported. (It is easy to check a number for a single factor.)
    2 KB (373 words) - 15:08, 5 June 2019
  • Residue
    ...umber is exactly divisible. For the L-L test a zero residue means that the number is [[prime]].
    1 KB (235 words) - 10:24, 6 February 2019
  • GIMPS clients
    So, to test a number efficiently, one must apply the theory to get the tests down to the "weeks" ...came popular among PC enthusiasts and [[Overclocking|overclockers]] as its number-crunching algorithms exercise the computer's processor and memory to their
    8 KB (1,218 words) - 15:37, 13 August 2020
  • Mfaktc
    ...ics cards, this is a very fast program. The name mfaktc is "'''M'''ersenne number '''fakt'''oring with '''C'''UDA", it is a mixture of English with the Germa
    5 KB (765 words) - 14:54, 25 February 2019
  • Mfakto
    ...akt'''oring with '''o'''penCL) is a port of ''[[mfaktc]]'' ('''m'''ersenne number '''fakt'''oring with '''C'''UDA) (for use on ATI/AMD GPUs rather than NVIDI -d <xy> use OpenCL platform number x and device number y
    17 KB (2,524 words) - 12:39, 24 January 2019
  • Rho factorization method
    The idea is to create a sequence iterating a polynomial modulo the number to be factored. ...nvented by Richard Brent in 1980 who used it to factor the eighth [[Fermat number]].
    3 KB (558 words) - 10:28, 6 February 2019
  • Riesel number
    ...value of ''k'' such that {{Kbn|k|n}} is always composite for all [[natural number]]s. In order to demonstrate whether 509203 is the smallest Riesel number or not (the '''[[Riesel problem 1]]'''), a [[distributed computing project]
    827 bytes (112 words) - 08:21, 25 March 2024
  • M12
    | number=170141183460...715884105727 ...d why this happened. Lucas was following a sequence (see [[Double Mersenne number]]). The first possible Mersenne prime (2<sup>1</sup>-1=2), when placed back
    2 KB (354 words) - 14:52, 19 September 2021
  • Primality test
    ...er a given number is [[prime]] or [[composite number|composite]]. When the number is declared composite, the algorithm does not reveal the prime [[factor]]s. ...(which is far slower than a probable primality test except when the input number has a special form) is run on it.
    3 KB (501 words) - 05:20, 3 August 2021
  • Lucas primality test
    ...ality test''' invented in 1891 by [[Édouard Lucas]], determines whether a number N is prime or not, using the complete factorization of N-1.
    1 KB (177 words) - 14:31, 17 February 2019
  • Square root
    ...e non-negative real number whose ''square'' (the result of multiplying the number by itself) is <math>x</math>. ...real numbers, the concept of [[imaginary number|imaginary]] and [[complex number]]s has been developed to provide a mathematical framework to deal with the
    13 KB (1,873 words) - 16:52, 24 October 2020
  • Primality testing program
    | [[Mersenne number]]s<br/>a &times; b<sup>n</sup>±c (only factoring and [[probable prime|PRP] | [[generalized Fermat number]]s
    2 KB (314 words) - 21:23, 29 August 2019
  • Pépin's test
    '''Pépin's test''' is mainly used for proving the primality of [[Fermat number]]s, but it is of no help for finding the factors of such numbers. ...for proving the primality of other numbers, like the [[Generalized Fermat number]]s <math>F_{n,2} = 4^{3^n}+2^{3^n}+1</math> with k = 5 instead of k = 3.
    2 KB (401 words) - 14:40, 6 March 2019
  • Proof by induction
    ...= F<sub>n-1</sub> + F<sub>n-2</sub> and <math>\phi </math> = 1.61803... a number such that <math>\phi^2 = \phi + 1</math>.
    4 KB (679 words) - 13:57, 20 February 2019

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