# A Swan-like note for a family of binary pentanomials

## TL;DR

This paper uses classical techniques to analyze the irreducibility of a specific family of binary pentanomials, revealing conditions under which these polynomials are reducible based on their degree modulo 8.

## Contribution

It extends Swan's method to a new family of binary pentanomials, providing criteria for their reducibility based on degree congruences.

## Key findings

- If n ≡ ±1 mod 8, the polynomial may be irreducible.
- If n ≠ ±1 mod 8, the polynomial is reducible.
- The paper applies classical techniques to a new polynomial family.

## Abstract

In this note, we employ the techniques of Swan (Pacific J. Math. 12(3): 1099-1106, 1962) with the purpose of studying the parity of the number of the irreducible factors of the penatomial $X^n+X^{3s}+X^{2s}+X^{s}+1\in\mathbb{F}_2[X]$, where $s$ is even and $n>3s$. Our results imply that if $n \not\equiv \pm 1 \pmod{8}$, then the polynomial in question is reducible.