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arXiv · 1908.00106

On (unitary) perfect polynomials over $\mathbb{F}_2$ with only Mersenne primes as odd divisors

Luis H. Gallardo·Olivier Rahavandrainy
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Abstract

The only (unitary) perfect polynomials over $\mathbb{F}_2$ that are products of $x$, $x+1$ and 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 $h$. Other consequences of such a factorization are new results about odd perfect polynomials.

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Luis H. Gallardo, Olivier Rahavandrainy. 2019-07-31. On (unitary) perfect polynomials over $\mathbb{F}_2$ with only Mersenne primes as odd divisors. https://arxiv.org/abs/1908.00106

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