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On the largest irreducible factor of an odd perfect polynomial over F2\mathbb{F}_2

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

A polynomial AA over F2\mathbb{F}_2 is perfect if it equals the sum σ(A)\sigma(A) of all its divisors, and odd if it is coprime to x2+xx^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 (ω≥5\omega\ge 5, Gallardo and Rahavandrainy), its number of irreducible factors counted with multiplicity and its degree (Ω≥12\Omega\ge 12 and deg⁡A>200\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 bound a different invariant: every odd perfect polynomial A≠1A\ne 1 has an irreducible factor of degree at least 20. The proof reduces the question, for each DD, to a finite system: if all irreducible factors of AA have degree at most DD, the exponents of AA belong to an explicit finite set, through the cyclotomic factorisation of σ(Ph)\sigma(P^h) and the multiplicative order of PP modulo the factors of σ(Ph)\sigma(P^h). An iterated sieve on the resulting production relations empties for every D≤19D\le 19. The computation was carried out twice, by two independent programs, with identical results; the second program is attached as an ancillary file. As a cross-check of the method, the same programs reproduce, by the orbit argument of Gallardo and Rahavandrainy, the exclusion of every prime of degree at most 30 as the smallest factor of a special perfect polynomial, in agreement with the published bounds.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

References

  1. E. F. Canaday, The sum of the divisors of a polynomial , Duke Math. J. 8 (1941), 721--737.
  2. U. C. Cengiz, P. Pollack and E. Treviño, Counting perfect polynomials , Finite Fields Appl. 47 (2017), 242--255.
  3. L. H. Gallardo, On the equation ∏PΦ3(P)=∏PP2\prod_P\Phi_3(P)=\prod_P P^2 over $\F_2[x]$ , Gulf J. Math. 12 (2022), no. 2, 39--48.
  4. L. H. Gallardo and O. Rahavandrainy, Odd perfect polynomials over $\F_2$ , J. Théor. Nombres Bordeaux 19 (2007), 165--174.
  5. L. H. Gallardo and O. Rahavandrainy, There is no odd perfect polynomial over $\F_2$ with four prime factors , Port. Math. 66 (2009), 131--145.
  6. O. Rahavandrainy, On special perfect polynomials over $\F_2$ , arXiv:2606.08640, 2026.arXiv

Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 02, 2026