{"id":"hexagon/2610.00697","pdf":"https://hexagonmath.org/pdf/2610.00697v1","url":"https://hexagonmath.org/abs/2610.00697v1","tags":["cs.CC","msc:11Y11","msc:11Y16","msc:11T24","Jacobi sums","Kummer extensions","binomial congruences","cyclotomic rings","deterministic algorithms","polynomial time"],"score":0,"title":"Deterministic Polynomial-Time Primality Testing with a Single Binomial Congruence","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Joseph Shunia"],"reviews":0,"subject":"math.NT","summary":"We give a deterministic polynomial-time primality test whose main verification is the congruence $(1+X)^n=1+X^n$ in $C_n[X]/(X^r-b(Y))$, where $C_n=(\\mathbb{Z}/n\\mathbb{Z})[Y]/(\\Phi_r(Y))$. Two identities in the coefficient ring normalize $b$ and ensure compatibility with the main congruence. A finite-field root count and a lattice bound show that an accepted input is a prime power; elementary preprocessing excludes nontrivial perfect powers. We construct $b$ deterministically using Jacobi sums and the uniform Dedekind-zeta zero-free theorem of Cureton and Thorner, which extends OpenAI's…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-31","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}