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Divisibility conjecture for prime divisors of Mersenne numbers

Prove that for any odd prime p, if k is the least positive integer such that 2^k − 1 is divisible by p, then k divides p − 1, and demonstrate that such a k always exists.

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Background

From observed factorization patterns of Mersenne numbers M_n = 2n − 1, the article formulates a conjecture relating the minimal exponent k with p − 1. This conjecture is equivalent to Fermat’s theorem for base 2 and to the stated October 1640 broadening (that the exponents for which p divides 2n − 1 form a simple arithmetic progression whose first term divides p − 1).

While modern number theory proves the claim via the multiplicative group modulo p, the article presents it as a conjecture arising naturally from empirical data and Fermat’s broader perspective.

References

This leads to the conjecture that p always occurs in M_{p-1} (Fermat's theorem), and sometimes earlier, first in M_{k} for some divisor k of p-1. This is exactly Fermat's October statement for a=2, and we'll call the claim there, that the exponent k (the first occurrence of p as a divisor) must divide p-1, the divisibility conjecture, which implies Fermat's theorem (and vice versa).

— How did Fermat discover his theorem? (2502.11165 - Pengelley, 16 Feb 2025) in Subsection “The Divisibility Conjecture”