# Mersenne prime

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2n − 1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2p − 1 for some prime p.

Named after Marin Mersenne 51 Infinite Mersenne numbers 3, 7, 31, 127, 8191 282,589,933 − 1 (December 7, 2018) A000668Mersenne primes (of form 2^p - 1 where p is a prime)

The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043 in the OEIS) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (sequence A000668 in the OEIS).

Numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89.

Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality.

As of October 2020, 51 Mersenne primes are known. The largest known prime number, 282,589,933 − 1, is a Mersenne prime.[1] Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were checked at least once.[2]

Unsolved problem in mathematics:

Are there infinitely many Mersenne primes?

(more unsolved problems in mathematics)

Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes and predicts their order of growth. It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes congruent to 3 (mod 4). For these primes p, 2p + 1 (which is also prime) will divide Mp, for example, 23 | M11, 47 | M23, 167 | M83, 263 | M131, 359 | M179, 383 | M191, 479 | M239, and 503 | M251 (sequence A002515 in the OEIS). Since for these primes p, 2p + 1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p + 1, and the multiplicative order of 2 mod 2p + 1 must divide ${\displaystyle {\frac {(2p+1)-1}{2}}}$ = p. Since p is a prime, it must be p or 1. However, it cannot be 1 since ${\displaystyle \Phi _{1}(2)=1}$ and 1 has no prime factors, so it must be p. Hence, 2p + 1 divides ${\displaystyle \Phi _{p}(2)=2^{p}-1}$ and ${\displaystyle 2^{p}-1=M_{p}}$ cannot be prime.

The first four Mersenne primes are M2 = 3, M3 = 7, M5 = 31 and M7 = 127 and because the first Mersenne prime starts at M2, all Mersenne primes are congruent to 3 (mod 4). Other than M0 = 0 and M1 = 1, all other Mersenne numbers are also congruent to 3 (mod 4). Consequently, in the prime factorization of a Mersenne number (  M2 ) there must be at least one prime factor congruent to 3 (mod 4).

A basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity

{\displaystyle {\begin{aligned}2^{ab}-1&=(2^{a}-1)\cdot \left(1+2^{a}+2^{2a}+2^{3a}+\cdots +2^{(b-1)a}\right)\\&=(2^{b}-1)\cdot \left(1+2^{b}+2^{2b}+2^{3b}+\cdots +2^{(a-1)b}\right).\end{aligned}}}

This rules out primality for Mersenne numbers with composite exponent, such as M4 = 24 − 1 = 15 = 3 × 5 = (22 − 1) × (1 + 22).

Though the above examples might suggest that Mp is prime for all primes p, this is not the case, and the smallest counterexample is the Mersenne number

M11 = 211 − 1 = 2047 = 23 × 89.

The evidence at hand suggests that a randomly selected Mersenne number is much more likely to be prime than an arbitrary randomly selected odd integer of similar size.[3] Nonetheless, prime values of Mp appear to grow increasingly sparse as p increases. For example, eight of the first 11 primes p give rise to a Mersenne prime Mp (the correct terms on Mersenne's original list), while Mp is prime for only 43 of the first two million prime numbers (up to 32,452,843).

The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, since Mersenne numbers grow very rapidly. The Lucas–Lehmer primality test (LLT) is an efficient primality test that greatly aids this task, making it much easier to test the primality of Mersenne numbers than that of most other numbers of the same size. The search for the largest known prime has somewhat of a cult following. Consequently, a large amount of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing.

Arithmetic modulo a Mersenne number is particularly efficient on a binary computer, making them popular choices when a prime modulus is desired, such as the Park–Miller random number generator. To find a primitive polynomial of Mersenne number order requires knowing the factorization of that number, so Mersenne primes allow one to find to primitive polynomials of very high order. Such primitive trinomials are used in pseudorandom number generators with very large periods such as the Mersenne twister, generalized shift register and Lagged Fibonacci generators.

## Perfect numbers

Mersenne primes Mp are closely connected to perfect numbers. In the 4th century BC, Euclid proved that if 2p − 1 is prime, then 2p − 1(2p − 1) is a perfect number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form.[4] This is known as the Euclid–Euler theorem. It is unknown whether there are any odd perfect numbers.

## History

2 3 5 7 13 17 19 31 11 23 29 37 41 43 47 53 59 71 73 79 83 97 101 103 109 113 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 263 269 271 277 281 283 293 307 311 The first 64 prime exponents with those corresponding to Mersenne primes shaded in cyan and in bold, and those thought to do so by Mersenne in red and bold.

Mersenne primes take their name from the 17th-century French scholar Marin Mersenne, who compiled what was supposed to be a list of Mersenne primes with exponents up to 257. The exponents listed by Mersenne were as follows:

2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.

His list replicated the known primes of his time with exponents up to 19. His next entry, 31, was correct, but the list then became largely incorrect, as Mersenne mistakenly included M67 and M257 (which are composite) and omitted M61, M89, and M107 (which are prime). Mersenne gave little indication of how he came up with his list.[5]

Édouard Lucas proved in 1876 that M127 is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years until 1951, when Ferrier found a larger prime, ${\displaystyle (2^{148}+1)/17}$, using a desk calculating machine.[6]:page 22 M61 was determined to be prime in 1883 by Ivan Mikheevich Pervushin, though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number. This was the second-largest known prime number, and it remained so until 1911. Lucas had shown another error in Mersenne's list in 1876. Without finding a factor, Lucas demonstrated that M67 is actually composite. No factor was found until a famous talk by Frank Nelson Cole in 1903.[7] Without speaking a word, he went to a blackboard and raised 2 to the 67th power, then subtracted one. On the other side of the board, he multiplied 193,707,721 × 761,838,257,287 and got the same number, then returned to his seat (to applause) without speaking.[8] He later said that the result had taken him "three years of Sundays" to find.[9] A correct list of all Mersenne primes in this number range was completed and rigorously verified only about three centuries after Mersenne published his list.

## Searching for Mersenne primes

Fast algorithms for finding Mersenne primes are available, and as of June 2019 the seven largest known prime numbers are Mersenne primes.

The first four Mersenne primes M2 = 3, M3 = 7, M5 = 31 and M7 = 127 were known in antiquity. The fifth, M13 = 8191, was discovered anonymously before 1461; the next two (M17 and M19) were found by Pietro Cataldi in 1588. After nearly two centuries, M31 was verified to be prime by Leonhard Euler in 1772. The next (in historical, not numerical order) was M127, found by Édouard Lucas in 1876, then M61 by Ivan Mikheevich Pervushin in 1883. Two more (M89 and M107) were found early in the 20th century, by R. E. Powers in 1911 and 1914, respectively.

The most efficient method presently known for testing the primality of Mersenne numbers is the Lucas–Lehmer primality test. Specifically, it can be shown that for prime p > 2, Mp = 2p − 1 is prime if and only if Mp divides Sp − 2, where S0 = 4 and Sk = (Sk − 1)2 − 2 for k > 0.

During the era of manual calculation, all the exponents up to and including 257 were tested with the Lucas–Lehmer test and found to be composite. A notable contribution was made by retired Yale physics professor Horace Scudder Uhler, who did the calculations for exponents 157, 167, 193, 199, 227, and 229.[10] Unfortunately for those investigators, the interval they were testing contains the largest known relative gap between Mersenne primes: the next Mersenne prime exponent, 521, would turn out to be more than four times larger than the previous record of 127.

Graph of number of digits in largest known Mersenne prime by year – electronic era. The vertical scale is logarithmic in the number of digits, thus being a ${\displaystyle \log(\log(y))}$ function in the value of the prime.

The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. Alan Turing searched for them on the Manchester Mark 1 in 1949,[11] but the first successful identification of a Mersenne prime, M521, by this means was achieved at 10:00 pm on January 30, 1952 using the U.S. National Bureau of Standards Western Automatic Computer (SWAC) at the Institute for Numerical Analysis at the University of California, Los Angeles, under the direction of D. H. Lehmer, with a computer search program written and run by Prof. R. M. Robinson. It was the first Mersenne prime to be identified in thirty-eight years; the next one, M607, was found by the computer a little less than two hours later. Three more M1279, M2203, and M2281 — were found by the same program in the next several months. M4253 is the first Mersenne prime that is titanic, M44,497 is the first gigantic, and M6,972,593 was the first megaprime to be discovered, being a prime with at least 1,000,000 digits.[12] All three were the first known prime of any kind of that size. The number of digits in the decimal representation of Mn equals n × log102⌋ + 1, where x denotes the floor function (or equivalently ⌊log10Mn⌋ + 1).

In September 2008, mathematicians at UCLA participating in the Great Internet Mersenne Prime Search (GIMPS) won part of a $100,000 prize from the Electronic Frontier Foundation for their discovery of a very nearly 13-million-digit Mersenne prime. The prize, finally confirmed in October 2009, is for the first known prime with at least 10 million digits. The prime was found on a Dell OptiPlex 745 on August 23, 2008. This was the eighth Mersenne prime discovered at UCLA.[13] On April 12, 2009, a GIMPS server log reported that a 47th Mersenne prime had possibly been found. The find was first noticed on June 4, 2009, and verified a week later. The prime is 242,643,801 − 1. Although it is chronologically the 47th Mersenne prime to be discovered, it is smaller than the largest known at the time, which was the 45th to be discovered. On January 25, 2013, Curtis Cooper, a mathematician at the University of Central Missouri, discovered a 48th Mersenne prime, 257,885,161 − 1 (a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network.[14] On January 19, 2016, Cooper published his discovery of a 49th Mersenne prime, 274,207,281 − 1 (a number with 22,338,618 digits), as a result of a search executed by a GIMPS server network.[15][16][17] This was the fourth Mersenne prime discovered by Cooper and his team in the past ten years. On September 2, 2016, the Great Internet Mersenne Prime Search finished verifying all tests below M37,156,667, thus officially confirming its position as the 45th Mersenne prime.[18] On January 3, 2018, it was announced that Jonathan Pace, a 51-year-old electrical engineer living in Germantown, Tennessee, had found a 50th Mersenne prime, 277,232,917 − 1 (a number with 23,249,425 digits), as a result of a search executed by a GIMPS server network.[19] The discovery was made by a computer in the offices of a church in the same town.[20][21] On December 21, 2018, it was announced that The Great Internet Mersenne Prime Search (GIMPS) discovered the largest known prime number, 282,589,933 − 1, having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018.[22] In late 2020, GIMPS began using a new technique to rule out potential Mersenne primes called the Probable prime (PRP) test, based on development from Robert Gerbicz in 2017, and a simple way to verify tests developed by Krzysztof Pietrzak in 2018. Due to the low error rate and ease of proof, this nearly halved the computing time to rule out potential primes over the Lucas-Lehmer test (as two users would no longer have to perform the same test to confirm the other's result), although exponents passing the PRP test still require one to confirm their primality.[23] ## Theorems about Mersenne numbers 1. If a and p are natural numbers such that ap − 1 is prime, then a = 2 or p = 1. • Proof: a ≡ 1 (mod a − 1). Then ap ≡ 1 (mod a − 1), so ap − 1 ≡ 0 (mod a − 1). Thus a − 1 | ap − 1. However, ap − 1 is prime, so a − 1 = ap − 1 or a − 1 = ±1. In the former case, a = ap, hence a = 0, 1 (which is a contradiction, as neither −1 nor 0 is prime) or p = 1. In the latter case, a = 2 or a = 0. If a = 0, however, 0p − 1 = 0 − 1 = −1 which is not prime. Therefore, a = 2. 2. If 2p − 1 is prime, then p is prime. • Proof: Suppose that p is composite, hence can be written p = ab with a and b > 1. Then 2p − 1 = 2ab − 1 = (2a)b − 1 = (2a − 1)((2a)b−1 + (2a)b−2 + … + 2a + 1) so 2p − 1 is composite. By contrapositive, if 2p − 1 is prime then p is prime. 3. If p is an odd prime, then every prime q that divides 2p − 1 must be 1 plus a multiple of 2p. This holds even when 2p − 1 is prime. • For example, 25 − 1 = 31 is prime, and 31 = 1 + 3 × (2 × 5). A composite example is 211 − 1 = 23 × 89, where 23 = 1 + (2 × 11) and 89 = 1 + 4 × (2 × 11). • Proof: By Fermat's little theorem, q is a factor of 2q−1 − 1. Since q is a factor of 2p − 1, for all positive integers c, q is also a factor of 2pc − 1. Since p is prime and q is not a factor of 21 − 1, p is also the smallest positive integer x such that q is a factor of 2x − 1. As a result, for all positive integers x, q is a factor of 2x − 1 if and only if p is a factor of x. Therefore, since q is a factor of 2q−1 − 1, p is a factor of q − 1 so q ≡ 1 (mod p). Furthermore, since q is a factor of 2p − 1, which is odd, q is odd. Therefore, q ≡ 1 (mod 2p). • This fact leads to a proof of Euclid's theorem, which asserts the infinitude of primes, distinct from the proof written by Euclid: for every odd prime p, all primes dividing 2p − 1 are larger than p; thus there are always larger primes than any particular prime. • It follows from this fact that for every prime p > 2, there is at least one prime of the form 2kp+1 less than or equal to Mp, for some integer k. 4. If p is an odd prime, then every prime q that divides 2p − 1 is congruent to ±1 (mod 8). • Proof: 2p+1 ≡ 2 (mod q), so 21/2(p+1) is a square root of 2 mod q. By quadratic reciprocity, every prime modulus in which the number 2 has a square root is congruent to ±1 (mod 8). 5. A Mersenne prime cannot be a Wieferich prime. • Proof: We show if p = 2m − 1 is a Mersenne prime, then the congruence 2p−1 ≡ 1 (mod p2) does not hold. By Fermat's little theorem, m | p − 1. Therefore, one can write p − 1 = . If the given congruence is satisfied, then p2 | 2 − 1, therefore 0 ≡ 2 − 1/2m − 1 = 1 + 2m + 22m + ... + 2(λ − 1)m ≡ −λ mod (2m − 1). Hence 2m − 1 | λ, and therefore λ ≥ 2m − 1. This leads to p − 1 ≥ m(2m − 1), which is impossible since m ≥ 2. 6. If m and n are natural numbers then m and n are coprime if and only if 2m − 1 and 2n − 1 are coprime. Consequently, a prime number divides at most one prime-exponent Mersenne number.[24] That is, the set of pernicious Mersenne numbers is pairwise coprime. 7. If p and 2p + 1 are both prime (meaning that p is a Sophie Germain prime), and p is congruent to 3 (mod 4), then 2p + 1 divides 2p − 1.[25] • Example: 11 and 23 are both prime, and 11 = 2 × 4 + 3, so 23 divides 211 − 1. • Proof: Let q be 2p + 1. By Fermat's little theorem, 22p ≡ 1 (mod q), so either 2p ≡ 1 (mod q) or 2p ≡ −1 (mod q). Supposing latter true, then 2p+1 = (21/2(p + 1))2 ≡ −2 (mod q), so −2 would be a quadratic residue mod q. However, since p is congruent to 3 (mod 4), q is congruent to 7 (mod 8) and therefore 2 is a quadratic residue mod q. Also since q is congruent to 3 (mod 4), −1 is a quadratic nonresidue mod q, so −2 is the product of a residue and a nonresidue and hence it is a nonresidue, which is a contradiction. Hence, the former congruence must be true and 2p + 1 divides Mp. 8. All composite divisors of prime-exponent Mersenne numbers are strong pseudoprimes to the base 2. 9. With the exception of 1, a Mersenne number cannot be a perfect power. That is, and in accordance with Mihăilescu's theorem, the equation 2m − 1 = nk has no solutions where m, n, and k are integers with m > 1 and k > 1. ## List of known Mersenne primes The table below lists all known Mersenne primes (sequence A000043 (p) and A000668 (Mp) in OEIS): # p Mp Mp digits Discovered Discoverer Method used 1 2 3 1 c. 430 BC Ancient Greek mathematicians[26] 2 3 7 1 c. 430 BC Ancient Greek mathematicians[26] 3 5 31 2 c. 300 BC Ancient Greek mathematicians[27] 4 7 127 3 c. 300 BC Ancient Greek mathematicians[27] 5 13 8191 4 1456[28] Anonymous[29][30][28] Trial division 6 17 131071 6 1588[31][28] Pietro Cataldi[28] Trial division[32] 7 19 524287 6 1588[28] Pietro Cataldi[28] Trial division[33] 8 31 2147483647 10 1772 Leonhard Euler[34][35] Trial division with modular restrictions[36] 9 61 2305843009213693951 19 1883 November[37] Ivan M. Pervushin Lucas sequences 10 89 618970019642...137449562111 27 1911 June[38] Ralph Ernest Powers Lucas sequences 11 107 162259276829...578010288127 33 1914 June 1[39][40][41] Ralph Ernest Powers[42] Lucas sequences 12 127 170141183460...715884105727 39 1876 January 10[43] Édouard Lucas Lucas sequences 13 521 686479766013...291115057151 157 1952 January 30[44] Raphael M. Robinson LLT / SWAC 14 607 531137992816...219031728127 183 1952 January 30[44] Raphael M. Robinson LLT / SWAC 15 1,279 104079321946...703168729087 386 1952 June 25[45] Raphael M. Robinson LLT / SWAC 16 2,203 147597991521...686697771007 664 1952 October 7[46] Raphael M. Robinson LLT / SWAC 17 2,281 446087557183...418132836351 687 1952 October 9[46] Raphael M. Robinson LLT / SWAC 18 3,217 259117086013...362909315071 969 1957 September 8[47] Hans Riesel LLT / BESK 19 4,253 190797007524...815350484991 1,281 1961 November 3[48][49] Alexander Hurwitz LLT / IBM 7090 20 4,423 285542542228...902608580607 1,332 1961 November 3[48][49] Alexander Hurwitz LLT / IBM 7090 21 9,689 478220278805...826225754111 2,917 1963 May 11[50] Donald B. Gillies LLT / ILLIAC II 22 9,941 346088282490...883789463551 2,993 1963 May 16[50] Donald B. Gillies LLT / ILLIAC II 23 11,213 281411201369...087696392191 3,376 1963 June 2[50] Donald B. Gillies LLT / ILLIAC II 24 19,937 431542479738...030968041471 6,002 1971 March 4[51] Bryant Tuckerman LLT / IBM 360/91 25 21,701 448679166119...353511882751 6,533 1978 October 30[52] Landon Curt Noll & Laura Nickel LLT / CDC Cyber 174 26 23,209 402874115778...523779264511 6,987 1979 February 9[52] Landon Curt Noll LLT / CDC Cyber 174 27 44,497 854509824303...961011228671 13,395 1979 April 8[53][54] Harry L. Nelson & David Slowinski LLT / Cray 1 28 86,243 536927995502...709433438207 25,962 1982 September 25 David Slowinski LLT / Cray 1 29 110,503 521928313341...083465515007 33,265 1988 January 29[55][56] Walter Colquitt & Luke Welsh LLT / NEC SX-2[57] 30 132,049 512740276269...455730061311 39,751 1983 September 19[58] David Slowinski LLT / Cray X-MP 31 216,091 746093103064...103815528447 65,050 1985 September 1[59][60] David Slowinski LLT / Cray X-MP/24 32 756,839 174135906820...328544677887 227,832 1992 February 17 David Slowinski & Paul Gage LLT / Harwell Lab's Cray-2[61] 33 859,433 129498125604...243500142591 258,716 1994 January 4[62][63][64] David Slowinski & Paul Gage LLT / Cray C90 34 1,257,787 412245773621...976089366527 378,632 1996 September 3[65] David Slowinski & Paul Gage[66] LLT / Cray T94 35 1,398,269 814717564412...868451315711 420,921 1996 November 13 GIMPS / Joel Armengaud[67] LLT / Prime95 on 90 MHz Pentium 36 2,976,221 623340076248...743729201151 895,932 1997 August 24 GIMPS / Gordon Spence[68] LLT / Prime95 on 100 MHz Pentium 37 3,021,377 127411683030...973024694271 909,526 1998 January 27 GIMPS / Roland Clarkson[69] LLT / Prime95 on 200 MHz Pentium 38 6,972,593 437075744127...142924193791 2,098,960 1999 June 1 GIMPS / Nayan Hajratwala[70] LLT / Prime95 on 350 MHz Pentium II IBM Aptiva 39 13,466,917 924947738006...470256259071 4,053,946 2001 November 14 GIMPS / Michael Cameron[71] LLT / Prime95 on 800 MHz Athlon T-Bird 40 20,996,011 125976895450...762855682047 6,320,430 2003 November 17 GIMPS / Michael Shafer[72] LLT / Prime95 on 2 GHz Dell Dimension 41 24,036,583 299410429404...882733969407 7,235,733 2004 May 15 GIMPS / Josh Findley[73] LLT / Prime95 on 2.4 GHz Pentium 4 42 25,964,951 122164630061...280577077247 7,816,230 2005 February 18 GIMPS / Martin Nowak[74] LLT / Prime95 on 2.4 GHz Pentium 4 43 30,402,457 315416475618...411652943871 9,152,052 2005 December 15 GIMPS / Curtis Cooper & Steven Boone[75] LLT / Prime95 on 2 GHz Pentium 4 44 32,582,657 124575026015...154053967871 9,808,358 2006 September 4 GIMPS / Curtis Cooper & Steven Boone[76] LLT / Prime95 on 3 GHz Pentium 4 45 37,156,667 202254406890...022308220927 11,185,272 2008 September 6 GIMPS / Hans-Michael Elvenich[77] LLT / Prime95 on 2.83 GHz Core 2 Duo 46 42,643,801 169873516452...765562314751 12,837,064 2009 June 4[n 1] GIMPS / Odd M. Strindmo[78][n 2] LLT / Prime95 on 3 GHz Core 2 47 43,112,609 316470269330...166697152511 12,978,189 2008 August 23 GIMPS / Edson Smith[77] LLT / Prime95 on Dell Optiplex 745 57,000,000 17,158,709 Lowest unverified milestone[n 3] 48[n 4] 57,885,161 581887266232...071724285951 17,425,170 2013 January 25 GIMPS / Curtis Cooper[79] LLT / Prime95 on 3 GHz Intel Core2 Duo E8400[80] 49[n 4] 74,207,281 300376418084...391086436351 22,338,618 2016 January 7[n 5] GIMPS / Curtis Cooper[15] LLT / Prime95 on Intel Core i7-4790 50[n 4] 77,232,917 467333183359...069762179071 23,249,425 2017 December 26 GIMPS / Jon Pace[81] LLT / Prime95 on 3.3 GHz Intel Core i5-6600[82] 51[n 4] 82,589,933 148894445742...325217902591 24,862,048 2018 December 7 GIMPS / Patrick Laroche[1] LLT / Prime95 on Intel Core i5-4590T 105,000,000 31,608,149 Lowest unchecked milestone[n 3] 1. Although M42,643,801 was first reported by a machine on April 12, 2009, no human took notice of this fact until June 4, 2009. 2. Strindmo also uses the alias Stig M. Valstad. 3. As of 2021 July 21 according to GIMPS. 4. It is not verified whether any undiscovered Mersenne primes exist between the 47th (M43,112,609) and the 51st (M82,589,933) on this chart; the ranking is therefore provisional. 5. Although M74,207,281 was first reported by a machine on September 17, 2015, no human took notice of this fact until January 7, 2016. All Mersenne numbers below the 51st Mersenne prime (M82,589,933) have been tested at least once but some have not been double-checked. Primes are not always discovered in increasing order. For example, the 29th Mersenne prime was discovered after the 30th and the 31st. Similarly, M43,112,609 was followed by two smaller Mersenne primes, first 2 weeks later and then 9 months later.[83] M43,112,609 was the first discovered prime number with more than 10 million decimal digits. The largest known Mersenne prime (282,589,933 − 1) is also the largest known prime number.[1] The largest known prime has been a Mersenne prime since 1952, except between 1989 and 1992.[84] ## Factorization of composite Mersenne numbers Since they are prime numbers, Mersenne primes are divisible only by 1 and themselves. However, not all Mersenne numbers are Mersenne primes. Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. As of June 2019, 21,193 1 is the record-holder,[85] having been factored with a variant of the special number field sieve that allows the factorization of several numbers at once. See integer factorization records for links to more information. The special number field sieve can factorize numbers with more than one large factor. If a number has only one very large factor then other algorithms can factorize larger numbers by first finding small factors and then making a primality test on the cofactor. As of July 2021, the largest factorization with probable prime factors allowed is 210,443,557 1 = 37,289,325,994,807 × q, where q is a 3,143,811-digit probable prime. It was discovered by GIMPS participant with nickname "fre_games".[86] As of July 2021, the Mersenne number M1277 is the smallest composite Mersenne number with no known factors; it has no prime factors below 268.[87] The table below shows factorizations for the first 20 composite Mersenne numbers (sequence A244453 in the OEIS). p Mp Factorization of Mp 11 2047 23 × 89 23 8388607 47 × 178,481 29 536870911 233 × 1,103 × 2,089 37 137438953471 223 × 616,318,177 41 2199023255551 13,367 × 164,511,353 43 8796093022207 431 × 9,719 × 2,099,863 47 140737488355327 2,351 × 4,513 × 13,264,529 53 9007199254740991 6,361 × 69,431 × 20,394,401 59 57646075230343487 179,951 × 3,203,431,780,337 (13 digits) 67 147573952589676412927 193,707,721 × 761,838,257,287 (12 digits) 71 2361183241434822606847 228,479 × 48,544,121 × 212,885,833 73 9444732965739290427391 439 × 2,298,041 × 9,361,973,132,609 (13 digits) 79 604462909807314587353087 2,687 × 202,029,703 × 1,113,491,139,767 (13 digits) 83 967140655691...033397649407 167 × 57,912,614,113,275,649,087,721 (23 digits) 97 158456325028...187087900671 11,447 × 13,842,607,235,828,485,645,766,393 (26 digits) 101 253530120045...993406410751 7,432,339,208,719 (13 digits) × 341,117,531,003,194,129 (18 digits) 103 101412048018...973625643007 2,550,183,799 × 3,976,656,429,941,438,590,393 (22 digits) 109 649037107316...312041152511 745,988,807 × 870,035,986,098,720,987,332,873 (24 digits) 113 103845937170...992658440191 3,391 × 23,279 × 65,993 × 1,868,569 × 1,066,818,132,868,207 (16 digits) 131 272225893536...454145691647 263 × 10,350,794,431,055,162,386,718,619,237,468,234,569 (38 digits) The number of factors for the first 500 Mersenne numbers can be found at (sequence A046800 in the OEIS). ## Mersenne numbers in nature and elsewhere In the mathematical problem Tower of Hanoi, solving a puzzle with an n-disc tower requires Mn steps, assuming no mistakes are made.[88] The number of rice grains on the whole chessboard in the wheat and chessboard problem is M64. The asteroid with minor planet number 8191 is named 8191 Mersenne after Marin Mersenne, because 8191 is a Mersenne prime (3 Juno, 7 Iris, 31 Euphrosyne and 127 Johanna having been discovered and named during the 19th century).[89] In geometry, an integer right triangle that is primitive and has its even leg a power of 2 ( 4 ) generates a unique right triangle such that its inradius is always a Mersenne number. For example, if the even leg is 2n + 1 then because it is primitive it constrains the odd leg to be 4n 1, the hypotenuse to be 4n + 1 and its inradius to be 2n 1.[90] The Mersenne numbers were studied with respect to the total number of accepting paths of non-deterministic polynomial time Turing machines in 2018[91] and intriguing inclusions were discovered. ## Mersenne–Fermat primes A Mersenne–Fermat number is defined as 2pr 1/2pr 1 1, with p prime, r natural number, and can be written as MF(p, r). When r = 1, it is a Mersenne number. When p = 2, it is a Fermat number. The only known Mersenne–Fermat primes with r > 1 are MF(2, 2), MF(2, 3), MF(2, 4), MF(2, 5), MF(3, 2), MF(3, 3), MF(7, 2), and MF(59, 2).[92] In fact, MF(p, r) = Φpr(2), where Φ is the cyclotomic polynomial. ## Generalizations The simplest generalized Mersenne primes are prime numbers of the form f(2n), where f(x) is a low-degree polynomial with small integer coefficients.[93] An example is 264 − 232 + 1, in this case, n = 32, and f(x) = x2x + 1; another example is 2192 − 264 − 1, in this case, n = 64, and f(x) = x3x − 1. It is also natural to try to generalize primes of the form 2n − 1 to primes of the form bn − 1 (for b ≠ 2 and n > 1). However (see also theorems above), bn − 1 is always divisible by b − 1, so unless the latter is a unit, the former is not a prime. This can be remedied by allowing b to be an algebraic integer instead of an integer: ### Complex numbers In the ring of integers (on real numbers), if b − 1 is a unit, then b is either 2 or 0. But 2n − 1 are the usual Mersenne primes, and the formula 0n − 1 does not lead to anything interesting (since it is always −1 for all n > 0). Thus, we can regard a ring of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers. #### Gaussian Mersenne primes If we regard the ring of Gaussian integers, we get the case b = 1 + i and b = 1 − i, and can ask (WLOG) for which n the number (1 + i)n − 1 is a Gaussian prime which will then be called a Gaussian Mersenne prime.[94] (1 + i)n − 1 is a Gaussian prime for the following n: 2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289, 991961, 1203793, 1667321, 3704053, 4792057, ... (sequence A057429 in the OEIS) Like the sequence of exponents for usual Mersenne primes, this sequence contains only (rational) prime numbers. As for all Gaussian primes, the norms (that is, squares of absolute values) of these numbers are rational primes: 5, 13, 41, 113, 2113, 525313, 536903681, 140737471578113, ... (sequence A182300 in the OEIS). #### Eisenstein Mersenne primes One may encounter cases where such a Mersenne prime is also an Eisenstein prime, being of the form b = 1 + ω and b = 1 − ω. In these cases, such numbers are called Eisenstein Mersenne primes. (1 + ω)n − 1 is an Eisenstein prime for the following n: 2, 5, 7, 11, 17, 19, 79, 163, 193, 239, 317, 353, 659, 709, 1049, 1103, 1759, 2029, 5153, 7541, 9049, 10453, 23743, 255361, 534827, 2237561, ... (sequence A066408 in the OEIS) The norms (that is, squares of absolute values) of these Eisenstein primes are rational primes: 7, 271, 2269, 176419, 129159847, 1162320517, ... (sequence A066413 in the OEIS) #### Repunit primes The other way to deal with the fact that bn − 1 is always divisible by b − 1, it is to simply take out this factor and ask which values of n make ${\displaystyle {\frac {b^{n}-1}{b-1}}}$ be prime. (The integer b can be either positive or negative.) If, for example, we take b = 10, we get n values of: 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... (sequence A004023 in the OEIS), corresponding to primes 11, 1111111111111111111, 11111111111111111111111, ... (sequence A004022 in the OEIS). These primes are called repunit primes. Another example is when we take b = −12, we get n values of: 2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... (sequence A057178 in the OEIS), corresponding to primes −11, 19141, 57154490053, .... It is a conjecture that for every integer b which is not a perfect power, there are infinitely many values of n such that bn − 1/b − 1 is prime. (When b is a perfect power, it can be shown that there is at most one n value such that bn − 1/b − 1 is prime) Least n such that bn − 1/b − 1 is prime are (starting with b = 2, 0 if no such n exists) 2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, ... (sequence A084740 in the OEIS) For negative bases b, they are (starting with b = −2, 0 if no such n exists) 3, 2, 2, 5, 2, 3, 2, 3, 5, 5, 2, 3, 2, 3, 3, 7, 2, 17, 2, 3, 3, 11, 2, 3, 11, 0, 3, 7, 2, 109, 2, 5, 3, 11, 31, 5, 2, 3, 53, 17, 2, 5, 2, 103, 7, 5, 2, 7, 1153, 3, 7, 21943, 2, 3, 37, 53, 3, 17, 2, 7, 2, 3, 0, 19, 7, 3, 2, 11, 3, 5, 2, ... (sequence A084742 in the OEIS) (notice this OEIS sequence does not allow n = 2) Least base b such that bprime(n) − 1/b − 1 is prime are 2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, ... (sequence A066180 in the OEIS) For negative bases b, they are 3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, ... (sequence A103795 in the OEIS) #### Other generalized Mersenne primes Another generalized Mersenne number is ${\displaystyle {\frac {a^{n}-b^{n}}{a-b}}}$ with a, b any coprime integers, a > 1 and a < b < a. (Since anbn is always divisible by ab, the division is necessary for there to be any chance of finding prime numbers. In fact, this number is the same as the Lucas number Un(a + b, ab), since a and b are the roots of the quadratic equation x2 − (a + b)x + ab = 0, and this number equals 1 when n = 1) We can ask which n makes this number prime. It can be shown that such n must be primes themselves or equal to 4, and n can be 4 if and only if a + b = 1 and a2 + b2 is prime. (Since a4b4/ab = (a + b)(a2 + b2). Thus, in this case the pair (a, b) must be (x + 1, −x) and x2 + (x + 1)2 must be prime. That is, x must be in .) It is a conjecture that for any pair (a, b) such that for every natural number r > 1, a and b are not both perfect rth powers, and −4ab is not a perfect fourth power. there are infinitely many values of n such that anbn/ab is prime. (When a and b are both perfect rth powers for an r > 1 or when −4ab is a perfect fourth power, it can be shown that there are at most two n values with this property, since if so, then anbn/ab can be factored algebraically) However, this has not been proved for any single value of (a, b). For more information, see [95][96][97][98][99][100][101][102][103][104] a b numbers n such that anbn/ab is prime (some large terms are only probable primes, these n are checked up to 100000 for |b| ≤ 5 or |b| = a − 1, 20000 for 5 < |b| < a − 1) OEIS sequence 2 1 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, ..., 57885161, ..., 74207281, ..., 77232917, ..., 82589933, ... A000043 2 −1 3, 4*, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ..., 13347311, 13372531, ... A000978 3 2 2, 3, 5, 17, 29, 31, 53, 59, 101, 277, 647, 1061, 2381, 2833, 3613, 3853, 3929, 5297, 7417, 90217, 122219, 173191, 256199, 336353, 485977, 591827, 1059503, ... A057468 3 1 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, ... A028491 3 −1 2*, 3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, ... A007658 3 −2 3, 4*, 7, 11, 83, 149, 223, 599, 647, 1373, 8423, 149497, 388897, ... A057469 4 3 2, 3, 7, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993, 2671, 7547, 24019, 46301, 48121, 68597, 91283, 131497, 148663, 184463, 341233, ... A059801 4 1 2 (no others) 4 −1 2*, 3 (no others) 4 −3 3, 5, 19, 37, 173, 211, 227, 619, 977, 1237, 2437, 5741, 13463, 23929, 81223, 121271, ... A128066 5 4 3, 43, 59, 191, 223, 349, 563, 709, 743, 1663, 5471, 17707, 19609, 35449, 36697, 45259, 91493, 246497, 265007, 289937, ... A059802 5 3 13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, ... A121877 5 2 2, 5, 7, 13, 19, 37, 59, 67, 79, 307, 331, 599, 1301, 12263, 12589, 18443, 20149, 27983, ... A082182 5 1 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, ... A004061 5 −1 5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, ... A057171 5 −2 2*, 3, 17, 19, 47, 101, 1709, 2539, 5591, 6037, 8011, 19373, 26489, 27427, ... A082387 5 −3 2*, 3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789, ... A122853 5 −4 4*, 5, 7, 19, 29, 61, 137, 883, 1381, 1823, 5227, 25561, 29537, 300893, ... A128335 6 5 2, 5, 11, 13, 23, 61, 83, 421, 1039, 1511, 31237, 60413, 113177, 135647, 258413, ... A062572 6 1 2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, ... A004062 6 −1 2*, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, ... A057172 6 −5 3, 4*, 5, 17, 397, 409, 643, 1783, 2617, 4583, 8783, ... A128336 7 6 2, 3, 7, 29, 41, 67, 1327, 1399, 2027, 69371, 86689, 355039, ... A062573 7 5 3, 5, 7, 113, 397, 577, 7573, 14561, 58543, ... A128344 7 4 2, 5, 11, 61, 619, 2879, 2957, 24371, 69247, ... A213073 7 3 3, 7, 19, 109, 131, 607, 863, 2917, 5923, 12421, ... A128024 7 2 3, 7, 19, 79, 431, 1373, 1801, 2897, 46997, ... A215487 7 1 5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ... A004063 7 −1 3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, ... A057173 7 −2 2*, 5, 23, 73, 101, 401, 419, 457, 811, 1163, 1511, 8011, ... A125955 7 −3 3, 13, 31, 313, 3709, 7933, 14797, 30689, 38333, ... A128067 7 −4 2*, 3, 5, 19, 41, 47, 8231, 33931, 43781, 50833, 53719, 67211, ... A218373 7 −5 2*, 11, 31, 173, 271, 547, 1823, 2111, 5519, 7793, 22963, 41077, 49739, ... A128337 7 −6 3, 53, 83, 487, 743, ... A187805 8 7 7, 11, 17, 29, 31, 79, 113, 131, 139, 4357, 44029, 76213, 83663, 173687, 336419, 615997, ... A062574 8 5 2, 19, 1021, 5077, 34031, 46099, 65707, ... A128345 8 3 2, 3, 7, 19, 31, 67, 89, 9227, 43891, ... A128025 8 1 3 (no others) 8 −1 2* (no others) 8 −3 2*, 5, 163, 191, 229, 271, 733, 21059, 25237, ... A128068 8 −5 2*, 7, 19, 167, 173, 223, 281, 21647, ... A128338 8 −7 4*, 7, 13, 31, 43, 269, 353, 383, 619, 829, 877, 4957, 5711, 8317, 21739, 24029, 38299, ... A181141 9 8 2, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099, ... A059803 9 7 3, 5, 7, 4703, 30113, ... A273010 9 5 3, 11, 17, 173, 839, 971, 40867, 45821, ... A128346 9 4 2 (no others) 9 2 2, 3, 5, 13, 29, 37, 1021, 1399, 2137, 4493, 5521, ... A173718 9 1 (none) 9 −1 3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393, ... A057175 9 −2 2*, 3, 7, 127, 283, 883, 1523, 4001, ... A125956 9 −4 2*, 3, 5, 7, 11, 17, 19, 41, 53, 109, 167, 2207, 3623, 5059, 5471, 7949, 21211, 32993, 60251, ... A211409 9 −5 3, 5, 13, 17, 43, 127, 229, 277, 6043, 11131, 11821, ... A128339 9 −7 2*, 3, 107, 197, 2843, 3571, 4451, ..., 31517, ... A301369 9 −8 3, 7, 13, 19, 307, 619, 2089, 7297, 75571, 76103, 98897, ... A187819 10 9 2, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493, ... A062576 10 7 2, 31, 103, 617, 10253, 10691, ... A273403 10 3 2, 3, 5, 37, 599, 38393, 51431, ... A128026 10 1 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... A004023 10 −1 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ... A001562 10 −3 2*, 3, 19, 31, 101, 139, 167, 1097, 43151, 60703, 90499, ... A128069 10 −7 2*, 3, 5, 11, 19, 1259, 1399, 2539, 2843, 5857, 10589, ... 10 −9 4*, 7, 67, 73, 1091, 1483, 10937, ... A217095 11 10 3, 5, 19, 311, 317, 1129, 4253, 7699, 18199, 35153, 206081, ... A062577 11 9 5, 31, 271, 929, 2789, 4153, ... A273601 11 8 2, 7, 11, 17, 37, 521, 877, 2423, ... A273600 11 7 5, 19, 67, 107, 593, 757, 1801, 2243, 2383, 6043, 10181, 11383, 15629, ... A273599 11 6 2, 3, 11, 163, 191, 269, 1381, 1493, ... A273598 11 5 5, 41, 149, 229, 263, 739, 3457, 20269, 98221, ... A128347 11 4 3, 5, 11, 17, 71, 89, 827, 22307, 45893, 63521, ... A216181 11 3 3, 5, 19, 31, 367, 389, 431, 2179, 10667, 13103, 90397, ... A128027 11 2 2, 5, 11, 13, 331, 599, 18839, 23747, 24371, 29339, 32141, 67421, ... A210506 11 1 17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ... A005808 11 −1 5, 7, 179, 229, 439, 557, 6113, 223999, 327001, ... A057177 11 −2 3, 5, 17, 67, 83, 101, 1373, 6101, 12119, 61781, ... A125957 11 −3 3, 103, 271, 523, 23087, 69833, ... A128070 11 −4 2*, 7, 53, 67, 71, 443, 26497, ... A224501 11 −5 7, 11, 181, 421, 2297, 2797, 4129, 4139, 7151, 29033, ... A128340 11 −6 2*, 5, 7, 107, 383, 17359, 21929, 26393, ... 11 −7 7, 1163, 4007, 10159, ... 11 −8 2*, 3, 13, 31, 59, 131, 223, 227, 1523, ... 11 −9 2*, 3, 17, 41, 43, 59, 83, ... 11 −10 53, 421, 647, 1601, 35527, ... A185239 12 11 2, 3, 7, 89, 101, 293, 4463, 70067, ... A062578 12 7 2, 3, 7, 13, 47, 89, 139, 523, 1051, ... A273814 12 5 2, 3, 31, 41, 53, 101, 421, 1259, 4721, 45259, ... A128348 12 1 2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ... A004064 12 −1 2*, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... A057178 12 −5 2*, 3, 5, 13, 347, 977, 1091, 4861, 4967, 34679, ... A128341 12 −7 2*, 3, 7, 67, 79, 167, 953, 1493, 3389, 4871, ... 12 −11 47, 401, 509, 8609, ... A213216 *Note: if b < 0 and n is even, then the numbers n are not included in the corresponding OEIS sequence. A conjecture related to the generalized Mersenne primes:[3][105] (the conjecture predicts where is the next generalized Mersenne prime, if the conjecture is true, then there are infinitely many primes for all such (a,b) pairs) For any integers a and b which satisfy the conditions: 1. a > 1, a < b < a. 2. a and b are coprime. (thus, b cannot be 0) 3. For every natural number r > 1, a and b are not both perfect rth powers. (since when a and b are both perfect rth powers, it can be shown that there are at most two n value such that anbn/ab is prime, and these n values are r itself or a root of r, or 2) 4. −4ab is not a perfect fourth power (if so, then the number has aurifeuillean factorization). has prime numbers of the form ${\displaystyle R_{p}(a,b)={\frac {a^{p}-b^{p}}{a-b}}}$ for prime p, the prime numbers will be distributed near the best fit line ${\displaystyle Y=G\cdot \log _{a}(\log _{a}(R_{(a,b)}(n)))+C}$ where ${\displaystyle \lim _{n\rightarrow \infty }G={\frac {1}{e^{\gamma }}}=0.561459483566\ldots }$ and there are about ${\displaystyle (\log _{e}(N)+m\cdot \log _{e}(2)\cdot \log _{e}\left(\log _{e}(N))+{\frac {1}{\sqrt {N}}}-\delta \right)\cdot {\frac {e^{\gamma }}{\log _{e}(a)}}}$ prime numbers of this form less than N. • e is the base of the natural logarithm. • γ is the Euler–Mascheroni constant. • loga is the logarithm in base a. • R(a,b)(n) is the nth prime number of the form apbp/ab for prime p. • C is a data fit constant which varies with a and b. • δ is a data fit constant which varies with a and b. • m is the largest natural number such that a and b are both perfect 2m − 1th powers. We also have the following three properties: 1. The number of prime numbers of the form apbp/ab (with prime p) less than or equal to n is about eγ loga(loga(n)). 2. The expected number of prime numbers of the form apbp/ab with prime p between n and an is about eγ. 3. The probability that number of the form apbp/ab is prime (for prime p) is about eγ/p loge(a). If this conjecture is true, then for all such (a,b) pairs, let q be the nth prime of the form apbp/ab, the graph of loga(loga(q)) versus n is almost linear. (See [3]) When a = b + 1, it is (b + 1)nbn, a difference of two consecutive perfect nth powers, and if anbn is prime, then a must be b + 1, because it is divisible by ab. Least n such that (b + 1)nbn is prime are 2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, 54517, 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, ... (sequence A058013 in the OEIS) Least b such that (b + 1)prime(n)bprime(n) is prime are 1, 1, 1, 1, 5, 1, 1, 1, 5, 2, 1, 39, 6, 4, 12, 2, 2, 1, 6, 17, 46, 7, 5, 1, 25, 2, 41, 1, 12, 7, 1, 7, 327, 7, 8, 44, 26, 12, 75, 14, 51, 110, 4, 14, 49, 286, 15, 4, 39, 22, 109, 367, 22, 67, 27, 95, 80, 149, 2, 142, 3, 11, ... (sequence A222119 in the OEIS) ## See also • Repunit • Fermat prime • Power of two • Erdős–Borwein constant • Mersenne conjectures • Mersenne twister • Double Mersenne number • Prime95 / MPrime • Great Internet Mersenne Prime Search (GIMPS) • Largest known prime number • Titanic prime • Gigantic prime • Megaprime • Wieferich prime • Wagstaff prime • Cullen prime • Woodall prime • Proth prime • Solinas prime • Gillies' conjecture • Williams number ## References 1. "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018. 2. "GIMPS Milestones Report". Mersenne.org. Mersenne Research, Inc. Retrieved 5 December 2020. 3. Caldwell, Chris. "Heuristics: Deriving the Wagstaff Mersenne Conjecture". 4. Chris K. Caldwell, Mersenne Primes: History, Theorems and Lists 5. The Prime Pages, Mersenne's conjecture. 6. Hardy, G. H.; Wright, E. M. (1959). An Introduction to the Theory of Numbers (4th ed.). Oxford University Press. 7. Cole, F. N. (1 December 1903). "On the factoring of large numbers". Bulletin of the American Mathematical Society. 10 (3): 134–138. doi:10.1090/S0002-9904-1903-01079-9. 8. Bell, E.T. and Mathematical Association of America (1951). Mathematics, queen and servant of science. McGraw-Hill New York. p. 228. 9. "h2g2: Mersenne Numbers". BBC News. Archived from the original on December 5, 2014. 10. Horace S. Uhler (1952). "A Brief History of the Investigations on Mersenne Numbers and the Latest Immense Primes". Scripta Mathematica. 18: 122–131. 11. Brian Napper, The Mathematics Department and the Mark 1. 12. The Prime Pages, The Prime Glossary: megaprime. 13. Maugh II, Thomas H. (2008-09-27). "UCLA mathematicians discover a 13-million-digit prime number". Los Angeles Times. Retrieved 2011-05-21. 14. Tia Ghose. "Largest Prime Number Discovered". Scientific American. Retrieved 2013-02-07. 15. Cooper, Curtis (7 January 2016). "Mersenne Prime Number discovery – 274207281 1 is Prime!". Mersenne Research, Inc. Retrieved 22 January 2016. 16. Brook, Robert (January 19, 2016). "Prime number with 22 million digits is the biggest ever found". New Scientist. Retrieved 19 January 2016. 17. Chang, Kenneth (21 January 2016). "New Biggest Prime Number = 2 to the 74 Mil ... Uh, It's Big". The New York Times. Retrieved 22 January 2016. 18. "Milestones". Archived from the original on 2016-09-03. 19. "Mersenne Prime Discovery - 2^77232917-1 is Prime!". www.mersenne.org. Retrieved 2018-01-03. 20. "Largest-known prime number found on church computer". christianchronicle.org. January 12, 2018. 21. "Found: A Special, Mind-Bogglingly Large Prime Number". January 5, 2018. 22. "GIMPS Discovers Largest Known Prime Number: 2^82,589,933-1". Retrieved 2019-01-01. 23. "GIMPS - The Math - PrimeNet". www.mersenne.org. Retrieved 29 June 2021. 24. Will Edgington's Mersenne Page Archived 2014-10-14 at the Wayback Machine 25. Caldwell, Chris K. "Proof of a result of Euler and Lagrange on Mersenne Divisors". Prime Pages. 26. There is no mentioning among the ancient Egyptians of prime numbers, and they did not have any concept for prime numbers known today. In the Rhind papyrus (1650 BC) the Egyptian fraction expansions have fairly different forms for primes and composites, so it may be argued that they knew about prime numbers. "The Egyptians used ($) in the table above for the first primes r = 3, 5, 7, or 11 (also for r = 23). Here is another intriguing observation: That the Egyptians stopped the use of ($) at 11 suggests they understood (at least some parts of) Eratosthenes's Sieve 2000 years before Eratosthenes 'discovered' it." The Rhind 2/n Table [Retrieved 2012-11-11]. In the school of Pythagoras (b. about 570 – d. about 495 BC) and the Pythagoreans, we find the first sure observations of prime numbers. Hence the first two Mersenne primes, 3 and 7, were known to and may even be said to have been discovered by them. There is no reference, though, to their special form 22 1 and 23 1 as such. The sources to the knowledge of prime numbers among the Pythagoreans are late. The Neoplatonic philosopher Iamblichus, AD c. 245–c. 325, states that the Greek Platonic philosopher Speusippus, c. 408 – 339/8 BC, wrote a book named On Pythagorean Numbers. According to Iamblichus this book was based on the works of the Pythagorean Philolaus, c. 470–c. 385 BC, who lived a century after Pythagoras, 570 – c. 495 BC. In his Theology of Arithmetic in the chapter On the Decad, Iamblichus writes: "Speusippus, the son of Plato's sister Potone, and head of the Academy before Xenocrates, compiled a polished little book from the Pythagorean writings which were particularly valued at any time, and especially from the writings of Philolaus; he entitled the book On Pythagorean Numbers. In the first half of the book, he elegantly expounds linear numbers [that is, prime numbers], polygonal numbers and all sorts of plane numbers, solid numbers and the five figures which are assigned to the elements of the universe, discussing both their individual attributes and their shared features, and their proportionality and reciprocity." Iamblichus The Theology of Arithmetic translated by Robin Waterfiled, 1988, p. 112f. [Retrieved 2012-11-11]. Iamblichus also gives us a direct quote from Speusippus' book where Speusippus among other things writes: "Secondly, it is necessary for a perfect number [the concept "perfect number" is not used here in a modern sense] to contain an equal amount of prime and incomposite numbers, and secondary and composite numbers." Iamblichus The Theology of Arithmetic translated by Robin Waterfiled, 1988, p. 113. [Retrieved 2012-11-11]. For the Greek original text, see Speusippus of Athens: A Critical Study with a Collection of the Related Texts and Commentary by Leonardo Tarán, 1981, p. 140 line 21–22 [Retrieved 2012-11-11] In his comments to Nicomachus of Gerasas's Introduction to Arithmetic, Iamblichus also mentions that Thymaridas, ca. 400 BC – ca. 350 BC, uses the term rectilinear for prime numbers, and that Theon of Smyrna, fl. AD 100, uses euthymetric and linear as alternative terms. Nicomachus of Gerasa, Introduction to Arithmetic, 1926, p. 127 [Retrieved 2012-11-11] It is unclear though when this said Thymaridas lived. "In a highly suspect passage in Iamblichus, Thymaridas is listed as a pupil of Pythagoras himself." Pythagoreanism [Retrieved 2012-11-11] Before Philolaus, c. 470–c. 385 BC, we have no proof of any knowledge of prime numbers. 27. "Euclid's Elements, Book IX, Proposition 36". 28. Arab mathematician Ismail ibn Ibrahim ibn Fallus (1194-1239) knew the first seven perfect numbers many years before they were discovered in Europe; see Perfect numbers from MacTutor History of Mathematics archive. Reference: Brentjes, Sonja (1987). "Die ersten sieben vollkommenen Zahlen und drei Arten befreundeter Zahlen in einem Werk zur elementaren Zahlentheorie von Ismā'īl b. Ibrāhīm b. Fallūs" [The first seven perfect numbers and three kinds of amicable numbers in a work on elementary number theory by Ismā'īl b. Ibrāhīm b. Fallūs]. NTM Schriftenreihe für Geschichte der Naturwissenschaften, Technik und Medizin (in German). 24 (1): 21–30. OCLC 812888599. Zbl 0625.01005.. 29. The Prime Pages, Mersenne Primes: History, Theorems and Lists. 30. We find the oldest (undisputed) note of the result in Codex nr. 14908, which origins from Bibliotheca monasterii ord. S. Benedicti ad S. Emmeramum Ratisbonensis now in the archive of the Bayerische Staatsbibliothek, see "Halm, Karl / Laubmann, Georg von / Meyer, Wilhelm: Catalogus codicum latinorum Bibliothecae Regiae Monacensis, Bd.: 2,2, Monachii, 1876, p. 250". [retrieved on 2012-09-17] The Codex nr. 14908 consists of 10 different medieval works on mathematics and related subjects. The authors of most of these writings are known. Some authors consider the monk Fridericus Gerhart (Amman), 1400–1465 (Frater Fridericus Gerhart monachus ordinis sancti Benedicti astrologus professus in monasterio sancti Emmerani diocesis Ratisponensis et in ciuitate eiusdem) to be the author of the part where the prime number 8191 is mentioned. Geschichte Der Mathematik [retrieved on 2012-09-17] The second manuscript of Codex nr. 14908 has the name "Regulae et exempla arithmetica, algebraica, geometrica" and the 5th perfect number and all is factors, including 8191, are mentioned on folio no. 34 a tergo (backside of p. 34). Parts of the manuscript have been published in Archiv der Mathematik und Physik, 13 (1895), pp. 388–406 [retrieved on 2012-09-23] 31. "A i lettori. Nel trattato de' numeri perfetti, che giàfino dell anno 1588 composi, oltrache se era passato auáti à trouarne molti auertite molte cose, se era anco amplamente dilatatala Tauola de' numeri composti , di ciascuno de' quali si vedeano per ordine li componenti, onde preposto unnum." p. 1 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#%5B%5D 32. pp. 13–18 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#%5B%5D 33. pp. 18–22 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#%5B%5D 34. http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv/1772&seite:int=36 Archived 2012-03-31 at the Wayback Machine Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres 1772, pp. 35–36 EULER, Leonhard: Extrait d'une lettre à M. Bernoulli, concernant le Mémoire imprimé parmi ceux de 1771. p. 318 [intitulé: Recherches sur les diviseurs de quelques nombres très grands compris dans la somme de la progression géométrique 1 + 101 + 102 + 103 + ... + 10T = S]. Retrieved 2011-10-02. 35. http://primes.utm.edu/notes/by_year.html#31 The date and year of discovery is unsure. Dates between 1752 and 1772 are possible. 36. Chris K. Caldwell. "Modular restrictions on Mersenne divisors". Primes.utm.edu. Retrieved 2011-05-21. 37. “En novembre de l’année 1883, dans la correspondance de notre Académie se trouve une communication qui contient l’assertion que le nombre 261 − 1 = 2305843009213693951 est un nombre premier. /…/ Le tome XLVIII des Mémoires Russes de l’Académie /…/ contient le compte-rendu de la séance du 20 décembre 1883, dans lequel l’objet de la communication du père Pervouchine est indiqué avec précision.” Bulletin de l'Académie Impériale des Sciences de St.-Pétersbourg, s. 3, v. 31, 1887, cols. 532–533. https://www.biodiversitylibrary.org/item/107789#page/277/mode/1up [retrieved 2012-09-17] See also Mélanges mathématiques et astronomiques tirés du Bulletin de l’Académie impériale des sciences de St.-Pétersbourg v. 6 (1881–1888), pp. 553–554. See also Mémoires de l'Académie impériale des sciences de St.-Pétersbourg: Sciences mathématiques, physiques et naturelles, vol. 48 38. Powers, R. E. (1 January 1911). "The Tenth Perfect Number". The American Mathematical Monthly. 18 (11): 195–197. doi:10.2307/2972574. JSTOR 2972574. 39. "M. E. Fauquenbergue a trouvé ses résultats depuis Février, et j’en ai reçu communication le 7 Juin; M. Powers a envoyé le 1er Juin un cablógramme à M. Bromwich [secretary of London Mathematical Society] pour M107. Sur ma demande, ces deux auteurs m’ont adressé leurs remarquables résultats, et je m’empresse de les publier dans nos colonnes, avec nos felicitations." p. 103, André Gérardin, Nombres de Mersenne pp. 85, 103–108 in Sphinx-Œdipe. [Journal mensuel de la curiosité, de concours & de mathématiques.] v. 9, No. 1, 1914. 40. "Power's cable announcing this same result was sent to the London Math. So. on 1 June 1914." Mersenne's Numbers, Scripta Mathematica, v. 3, 1935, pp. 112–119 http://primes.utm.edu/mersenne/LukeMirror/lit/lit_008s.htm [retrieved 2012-10-13] 41. "Records of Proceedings at Meetings". Proceedings of the London Mathematical Society. s2-13 (1): iv–xl. 1914. doi:10.1112/plms/s2-13.1.1-s. 42. The Prime Pages, M107: Fauquembergue or Powers?. 43. http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3039&I=166&M=chemindefer Presented at a meeting with Académie des sciences (France) on January 10, 1876. Retrieved 2011-10-02. 44. "Using the standard Lucas test for Mersenne primes as programmed by R. M. Robinson, the SWAC has discovered the primes 2521 1 and 2607 1 on January 30, 1952." D. H. Lehmer, Recent Discoveries of Large Primes, Mathematics of Computation, vol. 6, No. 37 (1952), p. 61, http://www.ams.org/journals/mcom/1952-06-037/S0025-5718-52-99404-0/S0025-5718-52-99404-0.pdf [Retrieved 2012-09-18] 45. "The program described in Note 131 (c) has produced the 15th Mersenne prime 21279 1 on June 25. The SWAC tests this number in 13 minutes and 25 seconds." D. H. Lehmer, A New Mersenne Prime, Mathematics of Computation, vol. 6, No. 39 (1952), p. 205, http://www.ams.org/journals/mcom/1952-06-039/S0025-5718-52-99387-3/S0025-5718-52-99387-3.pdf [Retrieved 2012-09-18] 46. "Two more Mersenne primes, 22203 1 and 22281 1, were discovered by the SWAC on October 7 and 9, 1952." D. H. Lehmer, Two New Mersenne Primes, Mathematics of Computation, vol. 7, No. 41 (1952), p. 72, http://www.ams.org/journals/mcom/1953-07-041/S0025-5718-53-99371-5/S0025-5718-53-99371-5.pdf [Retrieved 2012-09-18] 47. Riesel, Hans (1 January 1958). "A New Mersenne Prime". Mathematics of Computation. 12 (61): 60. doi:10.1090/S0025-5718-58-99282-2. 48. A. Hurwitz and J. L. Selfridge, Fermat numbers and perfect numbers, Notices of the American Mathematical Society, v. 8, 1961, p. 601, abstract 587-104. 49. "If p is prime, Mp = 2p − 1 is called a Mersenne number. The primes M4253 and M4423 were discovered by coding the Lucas-Lehmer test for the IBM 7090." Alexander Hurwitz, New Mersenne Primes, Mathematics of Computation, vol. 16, No. 78 (1962), pp. 249–251, http://www.ams.org/journals/mcom/1962-16-078/S0025-5718-1962-0146162-X/S0025-5718-1962-0146162-X.pdf [Retrieved 2012-09-18] 50. Gillies, Donald B. (1 January 1964). "Three new Mersenne primes and a statistical theory". Mathematics of Computation. 18 (85): 93–97. doi:10.1090/S0025-5718-1964-0159774-6. JSTOR 2003409. 51. Tuckerman, Bryant (1 October 1971). "The 24th Mersenne Prime". Proceedings of the National Academy of Sciences. 68 (10): 2319–2320. Bibcode:1971PNAS...68.2319T. doi:10.1073/pnas.68.10.2319. PMC 389411. PMID 16591945. 52. Noll, Curt; Nickel, Laura (October 1980). "The 25th and 26th Mersenne primes". Mathematics of Computation. 35 (152): 1387. doi:10.1090/S0025-5718-1980-0583517-4. JSTOR 2006405. 53. David Slowinski, "Searching for the 27th Mersenne Prime", Journal of Recreational Mathematics, v. 11(4), 1978–79, pp. 258–261, MR 80g #10013 54. "The 27th Mersenne prime. It has 13395 digits and equals 244497 1. [...] Its primeness was determined on April 8, 1979 using the Lucas–Lehmer test. The test was programmed on a CRAY-1 computer by David Slowinski & Harry Nelson." (p. 15) "The result was that after applying the Lucas–Lehmer test to about a thousand numbers, the code determined, on Sunday, April 8th, that 244497 1 is, in fact, the 27th Mersenne prime." (p. 17), David Slowinski, "Searching for the 27th Mersenne Prime", Cray Channels, vol. 4, no. 1, (1982), pp. 15–17. 55. Colquitt, W. N.; Welsh, L. (1 May 1991). "A new Mersenne prime". Mathematics of Computation. 56 (194): 867. Bibcode:1991MaCom..56..867C. doi:10.1090/S0025-5718-1991-1068823-9. JSTOR 2008415. 56. Peterson, I. (1988). "Priming for a Lucky Strike". Science News. 133 (6): 85. doi:10.2307/3972461. JSTOR 3972461. 57. "Mersenne Prime Numbers". Omes.uni-bielefeld.de. 2011-01-05. Retrieved 2011-05-21. 58. "Slowinski, a software engineer for Cray Research Inc. in Chippewa Falls, discovered the number at 11:36 a.m. Monday. [that is, 1983 September 19]" Jim Higgins, "Elusive numeral's number is up" and "Scientist finds big number" in The Milwaukee Sentinel – Sep 24, 1983, p. 1, p. 11 [retrieved 2012-10-23] 59. Peterson, I. (1985). "Prime Time for Supercomputers". Science News. 128 (13): 199. doi:10.2307/3970245. JSTOR 3970245. 60. "Slowinski's program also found the 28th in 1982, the 29th in 1983, and the 30th [known at that time] this past Labor Day weekend. [that is, August 31-September 1, 1985]" Rad Sallee, "`Supercomputer'/Chevron calculating device finds a bigger prime number" Houston Chronicle, Friday 09/20/1985, Section 1, Page 26, 4 Star Edition [retrieved 2012-10-23] 61. The Prime Pages, The finding of the 32nd Mersenne. 62. Chris Caldwell, The Largest Known Primes. 63. Crays press release 64. "Slowinskis email". 65. Silicon Graphics' press release https://web.archive.org/web/19970606011821/http://www.sgi.com/Headlines/1996/September/prime.html [Retrieved 2012-09-20] 66. The Prime Pages, A Prime of Record Size! 21257787 1. 67. Titanic Primes Raced to Win$100,000 Research Award. Retrieved on 2008-09-16.
68. "On April 12th [2009], the 47th known Mersenne prime, 242,643,801  1, a 12,837,064 digit number was found by Odd Magnar Strindmo from Melhus, Norway! This prime is the second largest known prime number, a "mere" 141,125 digits smaller than the Mersenne prime found last August.", The List of Largest Known Primes Home Page, http://primes.utm.edu/primes/page.php?id=88847 [retrieved 2012-09-18]
69. "GIMPS Discovers 48th Mersenne Prime, 257,885,161 1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. Retrieved 2016-01-19.
70. "List of known Mersenne prime numbers". Retrieved 29 November 2014.
71. "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". Mersenne Research, Inc. 3 January 2018. Retrieved 3 January 2018.
72. "List of known Mersenne prime numbers". Retrieved 3 January 2018.
73. GIMPS Milestones Report. Retrieved 2019-05-17
74. Caldwell, "The Largest Known Prime by Year: A Brief History" from the Prime Pages website, University of Tennessee at Martin.
75. Kleinjung, Thorsten; Bos, Joppe W.; Lenstra, Arjen K. (2014). "Mersenne Factorization Factory". Advances in Cryptology – ASIACRYPT 2014. Lecture Notes in Computer Science. 8874. pp. 358–377. doi:10.1007/978-3-662-45611-8_19. ISBN 978-3-662-45607-1.
76. Henri Lifchitz and Renaud Lifchitz. "PRP Top Records". Retrieved 2021-07-21.
77. "Exponent Status for M1277". Retrieved 2021-07-21.
78. Petković, Miodrag (2009). Famous Puzzles of Great Mathematicians. AMS Bookstore. p. 197. ISBN 978-0-8218-4814-2.
79. Alan Chamberlin. "JPL Small-Body Database Browser". Ssd.jpl.nasa.gov. Retrieved 2011-05-21.
80. "OEIS A016131". The On-Line Encyclopedia of Integer Sequences.
81. Tayfun Pay, and James L. Cox. "An overview of some semantic and syntactic complexity classes".
82. "A research of Mersenne and Fermat primes". Archived from the original on 2012-05-29.
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84. Chris Caldwell: The Prime Glossary: Gaussian Mersenne (part of the Prime Pages)
85. Zalnezhad, Ali; Zalnezhad, Hossein; Shabani, Ghasem; Zalnezhad, Mehdi (March 2015). "Relationships and Algorithm in order to Achieve the Largest Primes". arXiv:1503.07688 [math.NT].
86. (x, 1) and (x, −1) for x = 2 to 50
87. (x, 1) for x = 2 to 160
88. (x, −1) for x = 2 to 160
89. (x + 1, x) for x = 1 to 160
90. (x + 1, −x) for x = 1 to 40
91. (x + 2, x) for odd x = 1 to 107
92. (x, −1) for x = 2 to 200
93. PRP records, search for (a^n-b^n)/c, that is, (a, b)
94. PRP records, search for (a^n+b^n)/c, that is, (a, −b)
95. "Generalized Repunit Conjecture".