Number Theory


Conjecture on a Subset of Mersenne Numbers Divisible by Poulet Numbers

Authors: Marius Coman

The Poulet numbers (or the Fermat pseudoprimes to base 2) are defined by the fact that are the only composites n for which 2^(n – 1) – 1 is divisible by n (so, of course, all Mersenne numbers 2^(n - 1) – 1 are divisible by Poulet numbers if n is a Poulet number; but these are not the numbers I consider in this paper). In a previous paper I conjectured that any composite Mersenne number of the form 2^m – 1 with odd exponent m is divisible by a 2-Poulet number but seems that the conjecture was infirmed for m = 49. In this paper I conjecture that any Mersenne number (with even exponent) 2^(p – 1) – 1 is divisible by at least a Poulet number for any p prime, p ≥ 11, p ≠ 13.

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[v1] 2017-02-20 10:38:56

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