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arXiv · 2202.06357

All even (unitary) perfect polynomials over $\F_2$ with only Mersenne primes as odd divisors

Luis H. Gallardo·Olivier Rahavandrainy
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Abstract

We address an arithmetic problem in the ring $\F_2[x]$ related to the fixed points of the sum of divisors function. We study some binary polynomials $A$ such that $\sigma(A)/A $ is still a binary polynomial. Technically, we prove that the only (unitary) perfect polynomials over $\F_2$ that are products of $x$, $x+1$ and of Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1} +1$, for a Mersenne prime $M$ and for a positive integer $h$.

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Luis H. Gallardo, Olivier Rahavandrainy. 2022-02-13. All even (unitary) perfect polynomials over $\F_2$ with only Mersenne primes as odd divisors. https://arxiv.org/abs/2202.06357

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