# 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 thatpbe a prime of one of the forms 2^{2n}+1, 2^{2n}+/-3, 2^{2n+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**:

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

p= 2+/-1 or^{k}p= 4+/-3^{k}- 2
-1 is a prime (obviously a Mersenne prime)^{p}- (2
+1)/3 is a prime.^{p}

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. SelfridgeandWagstaff, 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.

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