were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 (sequence A109461 in the OEIS), and were composite for all other positive integers n ≤ 257.
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.
The New Mersenne conjecture or Bateman, Selfridge and Wagstaff conjecture (Bateman et al. 1989) states that for any odd natural number p, if any two of the following conditions hold, then so does the third: If p is an odd composite number, then 2p − 1 and (2p + 1)/3 are both composite.
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.
[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.
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.
[3][4] Lenstra, Pomerance, and Wagstaff have conjectured that there are infinitely many Mersenne primes, and, more precisely, that the number of Mersenne primes less than x is asymptotically approximated by where γ is the Euler–Mascheroni constant.
In other words, the number of Mersenne primes with exponent p less than y is asymptotically This means that there should on average be about
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,[8] rather than the expected number which is around 3.7.