The original, called Mersenne's conjecture, was a statement by Marin Mersenne in his Cogitata Physico-Mathematica (1644; see e.g. Dickson 1919) that the numbers were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integersn ≤ 257. The first seven entries of his list ( for n = 2, 3, 5, 7, 13, 17, 19) had already been proven to be primes by trial division before Mersenne's time;[1] only the last four entries were new claims by Mersenne. Due to the size of those last numbers, Mersenne did not and could not test all of them, nor could his peers in the 17th century. It was eventually determined, after three centuries and the availability of new techniques such as the Lucas–Lehmer test, that Mersenne's conjecture contained five errors, namely two entries are composite (those corresponding to the primes n = 67, 257) and three primes are missing (those corresponding to the primes n = 61, 89, 107). The correct list for n ≤ 257 is: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.
While Mersenne's original conjecture is false, it may have led to the New Mersenne conjecture.
New Mersenne conjecture
The New Mersenne conjecture or Bateman, Selfridge and Wagstaff conjecture (Bateman et al. 1989) states that for any oddnatural numberp, if any two of the following conditions hold, then so does the third:
p = 2k ± 1 or p = 4k ± 3 for some natural number k. (OEIS:A122834)
If p is an odd composite number, then 2p−1 and (2p+1)/3 are both composite. Therefore it is only necessary to test primes to verify the truth of the conjecture.
Currently, there are nine known numbers for which all three conditions hold: 3, 5, 7, 13, 17, 19, 31, 61, 127 (sequence A107360 in the OEIS). Bateman et al. expected that no number greater than 127 satisfies all three conditions, and showed that heuristically no greater number would even satisfy two conditions, which would make the New Mersenne conjecture trivially true.
If at least one of the double Mersenne numbers MM61 and MM127 is prime, then the New Mersenne conjecture would be false, since both M61 and M127 satisfy the first condition (since they are Mersenne primes themselves), but (2^M61+1)/3 and (2^M127+1)/3 are both composite, they are divisible by 1328165573307087715777 and 886407410000361345663448535540258622490179142922169401, respectively.
As of 2025[update], all the Mersenne primes up to 257885161 − 1 are known, and for none of these does the first condition or the third condition hold except for the ones just mentioned.[2][3][4][5] Primes which satisfy at least one condition are
Note that the two primes for which the original Mersenne conjecture is false (67 and 257) satisfy the first condition of the new conjecture (67 = 26 + 3, 257 = 28 + 1), but not the other two. 89 and 107, which were missed by Mersenne, satisfy the second condition but not the other two. Mersenne may have thought that 2p − 1 is prime only if p = 2k ± 1 or p = 4k ± 3 for some natural number k, but if he thought it was "if and only if" he would have included 61.
Status of new Mersenne conjecture for the first 100 primes
The New Mersenne conjecture can be thought of as an attempt to salvage the centuries-old Mersenne's conjecture, which is false. However, according to Robert D. Silverman, John Selfridge agreed that the New Mersenne conjecture is "obviously true" as it was chosen to fit the known data and counter-examples beyond those cases are exceedingly unlikely. It may be regarded more as a curious observation than as an open question in need of proving.
Prime Pages shows that the New Mersenne conjecture is true for all integers less than or equal to 10000000[2] by systematically listing all primes for which it is already known that one of the conditions holds. In fact, currently it is known whether the New Mersenne conjecture is true for all integers less than or equal to the current search limit of the Mersenne primes (see this page), see the list below: (currently it is still unknown the New Mersenne conjecture is true for the four integers 1073741827 = 2^30+3, 2147483647 = 2^31-1, 2305843009213693951 = 2^61-1, 170141183460469231731687303715884105727 = 2^127-1)
This means that there should on average be about ≈ 5.92 primes p of a given number of decimal digits such that is prime. The conjecture is fairly accurate for the first 40 Mersenne primes, but between 220,000,000 and 285,000,000 there are at least 12,[10] rather than the expected number which is around 3.7.
More generally, the number of primes p ≤ y such that is prime (where a, b are coprime integers, a > 1, −a < b < a, a and b are not both perfect r-th powers for any natural number r>1, and −4ab is not a perfect fourth power) is asymptotically
where m is the largest nonnegative integer such that a and −b are both perfect 2m-th powers. The case of Mersenne primes is one case of (a,b) = (2,1).
See also
Gillies' conjecture on the distribution of numbers of prime factors of Mersenne numbers
In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation:
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.
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.
In mathematics, a Fermat number, named after Pierre de Fermat (1607–1665), the first known to have studied them, is a positive integer of the form: where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ....
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy. Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called compositeness tests instead of primality tests.
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system. The word is a portmanteau of "repeated" and "digit". Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes.
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.
32 (thirty-two) is the natural number following 31 and preceding 33.
127 is the natural number following 126 and preceding 128. It is also a prime number.
In mathematics, a double Mersenne number is a Mersenne number of the form
In number theory, a Wagstaff prime is a prime number of the form
John Lewis Selfridge, was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics.
In mathematics, Wolstenholme's theorem states that for a prime number , the congruence
In number theory, a practical number or panarithmic number is a positive integer such that all smaller positive integers can be represented as sums of distinct divisors of . For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2.
In number theory, a Pierpont prime is a prime number of the form for some nonnegative integers u and v. That is, they are the prime numbers p for which p − 1 is 3-smooth. They are named after the mathematician James Pierpont, who used them to characterize the regular polygons that can be constructed using conic sections. The same characterization applies to polygons that can be constructed using ruler, compass, and angle trisector, or using paper folding.
The Bunyakovsky conjecture gives a criterion for a polynomial in one variable with integer coefficients to give infinitely many prime values in the sequence It was stated in 1857 by the Russian mathematician Viktor Bunyakovsky. The following three conditions are necessary for to have the desired prime-producing property:
The leading coefficient is positive,
The polynomial is irreducible over the rationals, and
There is no common factor for all the infinitely many values .
In number theory, Gillies' conjecture is a conjecture about the distribution of prime divisors of Mersenne numbers and was made by Donald B. Gillies in a 1964 paper in which he also announced the discovery of three new Mersenne primes. The conjecture is a specialization of the prime number theorem and is a refinement of conjectures due to I. J. Good and Daniel Shanks. The conjecture remains an open problem: several papers give empirical support, but it disagrees with the widely accepted Lenstra–Pomerance–Wagstaff conjecture.
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