{"id":"openai-math/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases","pdf":"https://github.com/openai/math/blob/main/preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026/simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf","url":"https://github.com/openai/math/tree/main/preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026","tags":["number-theory","primitive-roots","analytic-number-theory","sieve-methods"],"score":0,"title":"Simultaneous primitive roots: a conditional lower bound for prime bases","domain":"Number theory","family":"029","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"For every fixed finite set of distinct positive primes, we prove that at least $`cx/(\\log x)^2`$ primes in $`(x,2x)`$ have every member of the set as a primitive root, for some c > 0 and all sufficiently large x . The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-04","reactions":{},"formalized":false,"family_title":"Primitive roots for every admissible integer base"}