{"id":"hexagon/2611.00030","pdf":"https://hexagonmath.org/pdf/2611.00030v1","url":"https://hexagonmath.org/abs/2611.00030v1","tags":["msc:11T06","msc:11T55","msc:11Y16","Canaday conjecture","finite fields","perfect polynomials","sum of divisors"],"score":0,"title":"On the largest irreducible factor of an odd perfect polynomial over $\\mathbb{F}_2$","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["David Niedbala Giraudin"],"reviews":0,"subject":"math.NT","summary":"A polynomial $A$ over $\\mathbb{F}_2$ is perfect if it equals the sum $\\sigma(A)$ of all its divisors, and odd if it is coprime to $x^2+x$. Canaday conjectured in 1941 that 1 is the only odd perfect polynomial. Known necessary conditions on a counterexample bound its number of distinct irreducible factors ($\\omega\\ge 5$, Gallardo and Rahavandrainy), its number of irreducible factors counted with multiplicity and its degree ($\\Omega\\ge 12$ and $\\deg A>200$, Cengiz, Pollack and Trevino), and, when all exponents equal 2, the degree of its smallest irreducible factor (at least 40, Gallardo). We…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-11-03","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}