# There is no odd perfect polynomial over <b>F</b><sub>2</sub> with four prime factors

### Luis H. Gallardo

Université de Brest, France### Olivier Rahavandrainy

Université de Brest, France

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## Abstract

A perfect polynomial over the binary field **F**2 is a polynomial A ∈ **F**2[x] that equals the sum of all its divisors. If gcd(A,x2 + x) = 1 then we say that A is odd. It is believed that odd perfect polynomials do not exist. In this article we prove this for odd perfect polynomials A with four prime divisors, i.e., polynomials of the form A = PaQbRcSd where P, Q, R, S are distinct irreducible polynomials of degree > 1 over **F**2 and a, b, c, d are positive integers.