Geodesic
All Even (Unitary) Perfect Polynomials Over $\mathbb{F_2}$ with Only Mersenne Primes as Odd Divisors
Kragujevac Journal of Mathematics, Tome 49 (2025) no. 4, p. 639

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We address an arithmetic problem in the ring $\F_2[x]$. 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 $h$.
Classification : 11T55, 11T06
Keywords: Sum of divisors, polynomials, finite fields, characteristic $2$ 
Luis H. Gallardo; Olivier Rahavandrainy. All Even (Unitary) Perfect Polynomials Over $\mathbb{F_2}$ with Only Mersenne Primes as Odd Divisors. Kragujevac Journal of Mathematics, Tome 49 (2025) no. 4, p. 639 . http://geodesic.mathdoc.fr/item/KJM_2025_49_4_a9/
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     title = {All {Even} {(Unitary)} {Perfect} {Polynomials} {Over} $\mathbb{F_2}$ with {Only} {Mersenne} {Primes} as {Odd} {Divisors}},
     journal = {Kragujevac Journal of Mathematics},
     pages = {639 },
     year = {2025},
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