{"id":"hexagon/2610.00228","pdf":"https://hexagonmath.org/pdf/2610.00228v1","url":"https://hexagonmath.org/abs/2610.00228v1","tags":["msc:11R45","msc:11A07"],"score":0,"title":"Every nonsquare integer not equal to $-1$ is a primitive root modulo a positive lower density of primes","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Manjul Bhargava"],"reviews":0,"subject":"math.NT","summary":"For every integer \\(a\\) that is neither a square nor \\(-1\\), we prove that a positive lower density of primes have \\(a\\) as a primitive root. More precisely, a zero-free half-plane \\(\\Re s>1-1/N\\) for all Dedekind zeta functions gives a lower proportion of at least \\(c_a/(N\\log N)\\), with \\(c_a>0\\) independent of \\(N\\ge2\\). The theorem of Cureton and Thorner supplies \\(N=6000\\), so the positive-proportion conclusion is unconditional. The proof uses the marked Type II estimates of OpenAI to count primes $p$ such that $p-1$ is a small cofactor times one large prime. Allowing the cofactor to…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-14","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}