Mersenne's Conjecture
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Mersenne conjectures concern the characterization of a kind of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
s called
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 for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...
s, meaning prime numbers that are a
power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number 2, two as the Base (exponentiation), base and integer  as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^ ...
minus one.


Original Mersenne conjecture

The original, called Mersenne's conjecture, was a statement by
Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
in his ''Cogitata Physico-Mathematica'' (1644; see e.g. Dickson 1919) that the numbers 2^n - 1 were prime for ''n'' = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 , and were
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic material ...
for all other positive
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s ''n'' ≤ 257. The first seven entries of his list (2^n - 1 for ''n'' = 2, 3, 5, 7, 13, 17, 19) had already been proven to be primes by trial division before Mersenne's time; 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 In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
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 odd
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
''p'', if any two of the following conditions hold, then so does the third: # ''p'' = 2''k'' ± 1 or ''p'' = 4''k'' ± 3 for some natural number ''k''. () # 2''p'' − 1 is prime (a
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 for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...
). () # (2''p'' + 1)/3 is prime (a Wagstaff prime). () If ''p'' is an odd composite number, then 2''p'' − 1 and (2''p'' + 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 . 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 number In mathematics, a double Mersenne number is a Mersenne prime, Mersenne number of the form :M_ = 2^-1 where ''p'' is prime number, prime. Examples The first four terms of the integer sequence, sequence of double Mersenne numbers areChris Caldwe ...
s 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. , 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. Primes which satisfy at least one condition are :2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 67, 79, 89, 101, 107, 127, 167, 191, 199, 257, 313, 347, 521, 607, 701, 1021, 1279, 1709, 2203, 2281, 2617, 3217, 3539, 4093, 4099, 4253, 4423, 5807, 8191, 9689, 9941, ... 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 2''p'' − 1 is prime only if ''p'' = 2''k'' ± 1 or ''p'' = 4''k'' ± 3 for some natural number ''k'', but if he thought it was "
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
" he would have included 61. The New Mersenne conjecture can be thought of as an attempt to salvage the centuries-old Mersenne's conjecture, which is false. However, according t
Robert D. Silverman
John Selfridge John Lewis Selfridge (February 17, 1927 – October 31, 2010) was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics. Education Selfridge received his Ph.D. in ...
agreed that the New Mersenne conjecture is "obviously true" as it was chosen to fit the known data and
counter-example A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a co ...
s 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 The PrimePages is a website about prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is ...
shows that the New Mersenne conjecture is true for all integers less than or equal to 10000000 by systematically listing all primes for which it is already known that one of the conditions holds. In fact, currently it is known that the New Mersenne conjecture is true for all integers less than or equal to the current search limit of the Mersenne primes (se
this page
for the current search limit of the Mersenne primes), also currently it is known that the New Mersenne conjecture is true for all integers less than 1073741827 which satisfy the first condition, also currently it is known that the New Mersenne conjecture is true for all known integers which satisfy the second or third condition.


Lenstra–Pomerance–Wagstaff conjecture

Lenstra, Pomerance, and Wagstaff have conjectured that there are infinitely many
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 for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...
s, and, more precisely, that the number of Mersenne primes less than ''x'' is
asymptotically In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
approximated by :e^\gamma\cdot\log_2 \log_2(x),Heuristics: Deriving the Wagstaff Mersenne Conjecture
The Prime Pages. Retrieved on 2014-05-11.
where γ is the
Euler–Mascheroni constant Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
. In other words, the number of Mersenne primes with exponent ''p'' less than ''y'' is asymptotically :e^\gamma\cdot\log_2(y). This means that there should on average be about e^\gamma\cdot\log_2(10) ≈ 5.92 primes ''p'' of a given number of decimal digits such that M_p 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, rather than the expected number which is around 3.7. More generally, the number of primes ''p'' ≤ ''y'' such that \frac is prime (where ''a'', ''b'' are
coprime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
integers, ''a'' > 1, −''a'' < ''b'' < ''a'', ''a'' and ''b'' are not both perfect ''r''-th powers for any natural number ''r'' > 1, and −4''ab'' is not a perfect fourth power) is asymptotically :(e^\gamma+m\cdot\log_e(2))\cdot\log_a(y). where ''m'' is the largest nonnegative integer such that ''a'' and −''b'' are both perfect 2''m''-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 *
Lucas–Lehmer primality test In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1878 and subsequently proved by Derrick Henry Lehmer in 1930. The test The Lucas–Lehmer test works ...
* Lucas primality test * Catalan's Mersenne conjecture *
Mersenne's laws Mersenne's laws are laws describing the frequency of oscillation of a stretched string or monochord, useful in musical tuning and musical instrument construction. Overview The equation was first proposed by French mathematician and music theor ...


References

* * Reprinted by Chelsea Publishing, New York, 1971, .


External links

* The Prime Glossary
New Mersenne prime conjecture.
{{Prime number conjectures Conjectures about prime numbers Unsolved problems in number theory Mersenne primes