OPEN
This is open, and cannot be resolved with a finite computation.
Is it true that, for every prime $p$, there is a prime $q<p$ which is a primitive root modulo $p$?
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 $p$.
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Additional thanks to: Euro Vidal Sampaio
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T. F. Bloom, Erdős Problem #985, https://www.erdosproblems.com/985, accessed 2026-01-14
