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math.NT — Number Theory

Integrality of iterated averages and pointwise prime powers

Contributed by Darij Grinberg

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

A mix of human-written and AI-generated textHuman understanding: some partsInteger SequencesInteger-valued polynomialsdivisibilityfinite differencessupercongruences

Oct 29, 2026PDF ↗View work →

hexagon:2610.00589v1

math.NT — Number Theory

Nonconstant congruence speeds of integer tetration: period spectra, prime bases, explicit bounds, and densities

Contributed by Marco Ripà

Building on earlier human research into the constant congruence speed of integer tetration, including Appendix B of Ripà and Di Pietro (2026), and on a lifting-the-exponent (LTE) criterion publicly presented by Alper Ferudun for OEIS A390535 and A390576, this model-generated note proposes an explicit account of nonconstant speeds in non-squarefree numeral systems. Its principal results are an eventual floor-difference representation, an exact spectrum of attainable least periods, and explicit arithmetic progressions containing infinitely many prime bases whose congruence speed never becomes constant for each non-squarefree radix. The latter supplies a proposed proof of the coprime-infinitude assertion left conjectural in Appendix B of the earlier human-authored article. We also derive bounds for the least such base and exact natural densities, both among coprime bases and among all bases not divisible by the radical of the radix, together with relative prime densities. The mathematical derivations and manuscript are primarily GPT-6-generated. Ripà supplied the theoretical setting, earlier results and specialist guidance, and subsequently examined selected deductions and historical test cases. The manuscript has not received comprehensive independent human proof verification.

Primarily AI-generated textHuman understanding: some partscongruence speedeventual periodicityinteger tetrationlifting-the-exponent lemmanatural densitynon-squarefree radicesp-adic valuationsprimes in arithmetic progressions

Oct 29, 2026PDF ↗View work →

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Integrality of iterated averages and pointwise prime powers

Nonconstant congruence speeds of integer tetration: period spectra, prime bases, explicit bounds, and densities

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