New Mersenne Prime Conjecture

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  be a prime is that  be a prime of one of the forms , , ."

Mersenne's implication has been refuted, but Bateman, Selfridge, and Wagstaff (1989) 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 . If two of the following conditions hold, then so does the third:

1.  or ,

2.  is prime (a Mersenne prime),

3.  is prime (a Wagstaff prime).

This conjecture has been verified for all primes .

Based on the distribution and heuristics of (cf. https://www.utm.edu/research/primes/mersenne/heuristic.html) the known Mersenne and Wagstaff prime exponents, it seems quite likely that there is only a finite number of exponents satisfying the criteria of the new Mersenne conjecture. In fact, it is likely that there will be no more Mersenne or Wagstaff prime exponents discovered which fit the criteria. The new Mersenne conjecture may therefore simply be another instance of Guy's strong law of small numbers. In fact, R. D. Silverman (2005) has stated he was present when the conjecture was first posed and quotes Selfridge himself as describing the conjecture as a minor curious coincidence.

REFERENCES:

Bateman, P. T.; Selfridge, J. L.; and Wagstaff, S. S. "The New Mersenne Conjecture." Amer. Math. Monthly 96, 125-128, 1989.

Caldwell, C. "The New Mersenne Prime Conjecture." https://primes.utm.edu/mersenne/NewMersenneConjecture.html.

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, p. 28, 2005.

Silverman, R. D. Post to mersenneforum.org. Apr. 21, 2005. https://www.mersenneforum.org/showpost.php?p=53533&postcount=3.