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The invariant cMc_M of Gallardo and Rahavandrainy at Mersenne prime exponents, and the case p=7p=7

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

A Mersenne prime polynomial is an irreducible polynomial over F2\mathbb{F}_2 of the form xa(x+1)b+1x^a(x+1)^b+1 with a,b≥1a,b\ge1. Gallardo and Rahavandrainy conjectured that, apart from two exceptions, σ(M2h)\sigma(M^{2h}) always has an irreducible factor which is not a Mersenne prime. For a prime p≥5p\ge5 they proved it for every MM outside an exceptional set Σp=Σp1∪⋯∪Σp4\Sigma_p=\Sigma_p^1\cup\dots\cup\Sigma_p^4, defined through an integer cMc_M and an odd integer mMm_M read off the leading coefficients of σ(σ(Mp−1))\sigma(\sigma(M^{p-1})), and later they proved it for every Mersenne prime number p≥31p\ge31; their method does not reach p=7p=7. We compute cMc_M exactly when p=2d−1p=2^d-1 is a Mersenne prime number: cM=d+1c_M=d+1 if deg⁡M≠d\deg M\ne d, and cM=dc_M=d if deg⁡M=d\deg M=d. Consequently Σp1\Sigma_p^1 consists of the Mersenne primes of degree dd or d+1d+1, Σp2\Sigma_p^2 is empty, and Σp3\Sigma_p^3, Σp4\Sigma_p^4 are described by mM=deg⁡M−d−1m_M=\deg M-d-1 and mM=deg⁡M−dm_M=\deg M-d respectively. For p=7p=7 this reduces the conjecture to the Mersenne primes MM with mM∈{deg⁡M−4,deg⁡M−3}m_M\in\{\deg M-4,\deg M-3\}. A direct computation shows that σ(M6)\sigma(M^6) has a non-Mersenne irreducible factor for every Mersenne prime MM of degree at most 500500, and that mM≤17m_M\le17 in that range. Hence the conjecture holds whenever 77 divides 2h+12h+1 and deg⁡M≤500\deg M\le500, and a counterexample with 7∣2h+17\mid 2h+1 would need mM≥deg⁡M−4>496m_M\ge\deg M-4>496. A script attached as an ancillary file checks these computations, and reproduces the table of exceptional sets of Gallardo and Rahavandrainy.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

  1. L. H. Gallardo and O. Rahavandrainy, All even (unitary) perfect polynomials over $\F_2$ with only Mersenne primes as odd divisors , Kragujevac J. Math. 49 (2025), no. 4, 639--652.
  2. L. H. Gallardo and O. Rahavandrainy, Factorization of cyclotomic polynomial values at Mersenne prime polynomials , Mat. Vesnik, published online 2025, doi:10.57016/MV-HMLV9f07; arXiv:2106.10008.arXivDOI

Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 02, 2026