Mersenne prime

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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?

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 ${\textstyle {\frac {(2p+1)-1}{2}}=p}$ . 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 a 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 current 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 eight 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 as large as 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. M4,423 was the first prime discovered with more than 1000 digits, M44,497 was the first with more than 10,000, and M6,972,593 was the first with more than a million. In general, 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.[12]

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.[13]

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.[14][15][16] 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.[17]

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.[18] The discovery was made by a computer in the offices of a church in the same town.[19][20]

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.[21]

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.[22]

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.[23] 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.[24]
• 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

As of October 2021, the 51 known Mersenne primes are 2p − 1 for the following p:

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. (sequence A000043 in the OEIS)

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,[25] 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 running a primality test on the cofactor. As of April 2022, 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 a GIMPS participant with nickname "fre_games".[26] As of June 2022, the Mersenne number M1277 is the smallest composite Mersenne number with no known factors; it has no prime factors below 268,[27] and is very unlikely to have any factors below 1065 (~2216).[28]

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 576460752303423487 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.[29] 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).[30]

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.[31]

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).[32]

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.[33] 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.[34]

(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)

Divide an integer

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.)[a] 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.[b] It is a conjecture that for any pair (a, b) such that a and b are not both perfect rth powers for any r and −4ab is not a perfect fourth power, there are infinitely many values of n such that anbn/ab is prime.[c] However, this has not been proved for any single value of (a, b).

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.

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)

Notes

1. ^ 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.
2. ^ 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 .
3. ^ When a and b are both perfect rth powers for some r > 1 or when −4ab is a perfect fourth power, it can be shown that there are at most two values of n with this property: in these cases, anbn/ab can be factored algebraically.[citation needed]

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. ^ Maugh II, Thomas H. (2008-09-27). "UCLA mathematicians discover a 13-million-digit prime number". Los Angeles Times. Retrieved 2011-05-21.
13. ^ Tia Ghose. "Largest Prime Number Discovered". Scientific American. Retrieved 2013-02-07.
14. ^ Cooper, Curtis (7 January 2016). "Mersenne Prime Number discovery – 274207281 − 1 is Prime!". Mersenne Research, Inc. Retrieved 22 January 2016.
15. ^ Brook, Robert (January 19, 2016). "Prime number with 22 million digits is the biggest ever found". New Scientist. Retrieved 19 January 2016.
16. ^ 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.
17. ^ "Milestones". Archived from the original on 2016-09-03.
18. ^ "Mersenne Prime Discovery - 2^77232917-1 is Prime!". www.mersenne.org. Retrieved 2018-01-03.
19. ^ "Largest-known prime number found on church computer". christianchronicle.org. January 12, 2018.
20. ^ "Found: A Special, Mind-Bogglingly Large Prime Number". January 5, 2018.
21. ^ "GIMPS Discovers Largest Known Prime Number: 2^82,589,933-1". Retrieved 2019-01-01.
22. ^ "GIMPS - The Math - PrimeNet". www.mersenne.org. Retrieved 29 June 2021.
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24. ^ Caldwell, Chris K. "Proof of a result of Euler and Lagrange on Mersenne Divisors". Prime Pages.
25. ^ Kleinjung, Thorsten; Bos, Joppe W.; Lenstra, Arjen K. (2014). "Mersenne Factorization Factory". Advances in Cryptology – ASIACRYPT 2014. Lecture Notes in Computer Science. Vol. 8874. pp. 358–377. doi:10.1007/978-3-662-45611-8_19. ISBN 978-3-662-45607-1.
26. ^ Henri Lifchitz and Renaud Lifchitz. "PRP Top Records". Retrieved 2021-07-21.
27. ^ "Exponent Status for M1277". Retrieved 2021-07-21.
28. ^ "M1277 Mersenne number exponent details". www.mersenne.ca. Retrieved 24 June 2022.
29. ^ Petković, Miodrag (2009). Famous Puzzles of Great Mathematicians. AMS Bookstore. p. 197. ISBN 978-0-8218-4814-2.
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31. ^ "OEIS A016131". The On-Line Encyclopedia of Integer Sequences.
32. ^ "A research of Mersenne and Fermat primes". Archived from the original on 2012-05-29.
33. ^ Solinas, Jerome A. (1 January 2011). "Generalized Mersenne Prime". In Tilborg, Henk C. A. van; Jajodia, Sushil (eds.). Encyclopedia of Cryptography and Security. Springer US. pp. 509–510. doi:10.1007/978-1-4419-5906-5_32. ISBN 978-1-4419-5905-8.
34. ^ Chris Caldwell: The Prime Glossary: Gaussian Mersenne (part of the Prime Pages)
35. ^ 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].
36. ^ (x, 1) and (x, −1) for x = 2 to 50
37. ^ (x, 1) for x = 2 to 160
38. ^ (x, −1) for x = 2 to 160
39. ^ (x + 1, x) for x = 1 to 160
40. ^ (x + 1, −x) for x = 1 to 40
41. ^ (x + 2, x) for odd x = 1 to 107
42. ^ (x, −1) for x = 2 to 200
43. ^ PRP records, search for ${\displaystyle (a^{n}-b^{n})/c}$ , that is, (a, b)
44. ^ PRP records, search for ${\displaystyle (a^{n}+b^{n})/c}$ , that is, (a, −b)