All Even (Unitary) Perfect Polynomials Over $F_2$ with Only Mersenne Primes as Odd Divisors
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Authors: L. H. GALLARDO AND O. RAHAVANDRAINY
DOI: 10.46793/KgJMat2504.639G
Abstract:
We address an arithmetic problem in the ring F2[x]. We prove that
the only (unitary) perfect polynomials over F2 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
Keywords:
Sum of divisors, polynomials, finite fields, characteristic 2.
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Sci. Fis. Mat. Nat. 62 (1977), 417–422.
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(2019), 284–296. https://doi.org/10.1016/j.ffa.2019.06.006
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Paris 359(2) (2021), 123–130. https://doi.org/10.5802/crmath.149
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