Dickson states "In a letter to Tanner [L'intermediaire des math., 2, 1895, 317] Lucas stated that Mersenne (1644, 1647) implied that a necessary and sufficient condition that 2^p - 1 be a prime is that p be a prime of one of the forms 2^(2n) + 1, 2^(2n) ± 3, 2^(2n + 1) - 1." Mersenne's implication has been refuted, but Bateman, Selfridge, and Wagstaff used the statement as an inspiration for what is now called the new Mersenne conjecture, which can be stated as follows. Consider an odd natural number n. If two of the following conditions hold, then so does the third: 1.n = 2^k ± 1

Catalan-Mersenne number | Cunningham number | double Mersenne number | Fermat-Lucas number | integer sequence primes | Lucas-Lehmer test | Mersenne prime | perfect number | Wagstaff prime