Artin's primitive root conjecture

Artin conjectured in 1927 that every admissible integer is a primitive root modulo infinitely many primes.

Known results

• Hooley, 1967: proved the conjecture conditionally under the relevant Generalized Riemann Hypothesis.
• Gupta–Murty, 1984: proved it for infinitely many integers, without an explicit infinite list.
• Heath-Brown, 1986: showed that at most two prime values and at most three squarefree values can be exceptions; in particular, one of $2$, $3$, or $5$ works.
• No particular admissible integer is known unconditionally to work.

2026 claimed proof and new bounds

An undated version-$14$ preprint claims an unconditional positive-density asymptotic for every admissible integer, which would settle the conjecture, but supplies no independent verification or referee assessment. On August 25, 2026, a new paper by M. Ram Murty and Sunil Naik was announced on a non-abelian large sieve and Artin’s conjecture; the retrieved record gives no theorem details. A 2026 journal article by Chi Wa Chan reports uniform density bounds in quadratic number fields, but its precise advance cannot be assessed from the available record.

Current status (as of August 2026): Artin’s conjecture remains open unconditionally; Hooley’s conditional theorem and the classical partial results stand, while the claimed complete proof is unverified.