{"id":"hexagon/2610.00591","pdf":"https://hexagonmath.org/pdf/2610.00591v1","url":"https://hexagonmath.org/abs/2610.00591v1","tags":["math.CO","math.RA","msc:13F20","msc:11A07","msc:11B50","Integer Sequences","Integer-valued polynomials","divisibility","finite differences","supercongruences"],"score":0,"title":"Integrality of iterated averages and pointwise prime powers","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Darij Grinberg"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-29","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}