Fixed points of the sum of divisors function on ({{mathbb{F}}}_2[x])

We work on an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points (F) of the sum of divisors function (sigma : {mathbb{F}}_2[x] mapsto {mathbb{F}}_2[x]) (defined mutatis mutandi like the usual sum of divisors over the integers) of the form (F := A^2 cdot S), (S) square-free, with (omega(S) leq 3), coprime with (A), for (A) even, of whatever degree, under some conditions. This gives a characterization of (5) of the (11) known fixed points of (sigma) in ({mathbb{F}}_2[x])

Artin-Schreier, Erdős, and Kurepa\u27s conjecture

We discuss possible generalizations of Erdŏs\u27s problem about factorials in Fp to the Artin-Schreier extension Fpp of Fp. The generalizations are related to Bell numbers in Fp and to Kurepa\u27s conjecture

Under a mild condition, Ryser\u27s Conjecture holds for every ( n:= 4h^2) with h>1 odd and non square-free

We prove, under a mild condition, that there is no circulant Hadamard matrix ( H) with (n >4) rows when (sqrt{n/4}) is not square-free. The proof introduces a new method to attack Ryser\u27s Conjecture, that is a long standing difficult conjecture

On unitary splitting perfect polynomials over~ Fp2 ~F_{p^2}~

We classify some unitary splitting perfect polynomials over a finite field $F_{p^2}$, where $p$ is a prime number. This generalizes Beard\u27s work over $F_p$