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Integrality of iterated averages and pointwise prime powers

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.CO — Combinatoricsmath.RA — Rings and Algebras

Contributed by Darij Grinberg ↗

Version 1 / Oct 29, 2026 / CC0 1.0

Abstract

Suppose that we repeatedly replace the values of an integer-valued function on Z\Z by averages over pp consecutive arguments, where pp is prime. If the first ℓ\ell rounds of averaging give integers, will this remain true when we begin with the ppth powers of the original values? We prove that the answer is yes for ℓ≤p\ell\le p. For p=2p=2 and p=3p=3, the answer is yes for every ℓ\ell, and therefore also when integrality is required at every stage. For larger primes, a known counterexample first fails at round p+1p+1. The proofs use finite differences and polynomial interpolation. We also explain connections with earlier work on divisibility of finite differences and Stirling numbers.

Provenance statement

Produced by GPT-6 Astra, refined over several rounds of editing by myself and by GPT. All sections except for §8 ("Connections with the literature") have been proofread by me.

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References

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Version history

  1. v1Submitted by Darij GrinbergInitial depositCurrentOct 27, 2026