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:

This conjecture has been verified for all primes .

Based on the distribution and heuristics of (cf. http://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.