bibkey: hanna2016a271931
authors: Paul D. Hanna
year: 2016
title: "OEIS A271931, A271932 and A271933: prime substitution integrality"
doi: null
url: https://oeis.org/A271931
claim: "More generally, for prime p there exists an integer series G(x) that satisfies: G(x) = G( x^p + p*x*G(x)^p )^(1/p) with G(0)=0, G'(0)=1 (conjecture)."
strata_touched:
  - D5/S1/Recurrence/Invariants/HannaPrimeSubstitutionIntegrality
license: citation-only
triage: anchor

OEIS A271931, A271932 and A271933: prime substitution integrality

Verified locator

Source: https://oeis.org/A271931

Statement

More generally, for prime p there exists an integer series G(x) that satisfies: G(x) = G( x^p + pxG(x)^p )^(1/p) with G(0)=0, G’(0)=1 (conjecture).

The same comment appears in A271932 and A271933. The entries are unpaginated. Their exponents are 5, 7 and 11, and the comment quantifies over all primes. Integer series is encoded as PowerSeries ℤ; normalization is constantCoeff G = 0 and coeff 1 G = 1. The root with linear coefficient one is expressed by G^p = G(X^p + p*X*G^p). The proof constructs coefficients directly: modulo p the residual vanishes by Frobenius, so each correction is integral. The construction also works with an integer c divisible by p in place of p in the substitution argument.

Literature-screen completeness is ASSUMED-UNVERIFIED. The source entries cite no proof of these conjectures; this is a statement about the checked entries, not a claim of exhaustive literature coverage.