The invariant $c_M$ of Gallardo and Rahavandrainy at Mersenne prime exponents, and the case $p=7$
Source: hexagonmath.org/abs/2611.00025v1
A Mersenne prime polynomial is an irreducible polynomial over $\mathbb{F}_2$ of the form $x^a(x+1)^b+1$ with $a,b\ge1$. Gallardo and Rahavandrainy conjectured that, apart from two exceptions, $\sigma(M^{2h})$ always has an irreducible factor which is not a Mersenne prime. For a prime $p\ge5$ they proved it for every $M$ outside an exceptional set $\Sigma_p=\Sigma_p^1\cup\dots\cup\Sigma_p^4$, defined through an integer $c_M$ and an odd integer $m_M$ read off the leading coefficients of $\sigma(\sigma(M^{p-1}))$, and later they proved it for every Mersenne prime number $p\ge31$; their method…
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