Problem Statement
Is it true that, for every prime $p$, there is a prime $q<p$ which is a primitive root modulo $p$?
Categories:
Number Theory
Progress
Artin conjectured that $2$ is a primitive root for infinitely many primes $p$, which Hooley [Ho67b] proved assuming the Generalised Riemann Hypothesis. Heath-Brown [He86b] proved that at least one of $2$, $3$, or $5$ is a primitive root for infinitely many primes $p$.Source: erdosproblems.com/985 | Last verified: January 19, 2026