new mersenne prime conjecture

In our entry on Mersenne's conjecture we give a longer version of the following quote from Dickson's history [Dickson19]:

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 2²ⁿ+1, 2²ⁿ+/-3, 2²ⁿ⁺¹-1.

In that entry we point out these conditions are neither necessary nor sufficient. So is there anything that can be said in this regard? Bateman, Selfridge, and Wagstaff say yes and have made the The New Mersenne Conjecture:

Let p be any odd natural number. If two of the following conditions hold, then so does the third:

p = 2^k+/-1 or p = 4^k+/-3

2^p-1 is a prime (obviously a Mersenne prime)

(2^p+1)/3 is a prime.

This conjecture has been verified for all primes p less than 100000, and for all known Mersenne primes. Some feel that "conjecture" is too strong of a word for the above and that perhaps this is even another case of Guy's law of small numbers.

See Also: Mersennes

Related pages (outside of this work)

Status of the New Mersenne Prime Conjecture Originally by Conrad Curry
Status of the New Mersenne Prime Conjecture by Renaud Lifchitz

References:

BSW89
P. T. Bateman, J. L. Selfridge and Wagstaff, Jr., S. S., "The new Mersenne conjecture," Amer. Math. Monthly, 96 (1989) 125-128. MR 90c:11009
Dickson19 (Vol. 1, p28)
L. E. Dickson, History of the theory of numbers, Carnegie Institute of Washington, 1919. Reprinted by Chelsea Publishing, New York, 1971.