erdős #985
Is it true that, for every prime , there is a prime which is a primitive root modulo ?
Worked, still open.
number theory · open · formalized (Lean) · 0 attempts
machinery: artin-primitive-root,least-primitive-root,GRH-conditional,character-sum-Burgess,prime-distribution,sieve/Brun-Titchmarsh,Hooley-Heath-Brown-density
use this record
vela registry pull vfr_37aec80d874a0239vela reproduce examples/erdos-problemsevidence
unverified AI candidates (2)
gpt-erdos · GPT-5.2 Pro + Deep Research · unverified
Taken literally, the statement is **false** because of the trivial case $p=2$: there is no prime (q<p).
candidate solution ↗llm-hunter · gpt pro 5.2 · unverified
1 LLM attack(s) recorded (gpt pro 5.2); unverified.
candidate solution ↗formal
AMS 11 · open (literature)
theorem erdos_985 : answer(sorry) ↔ ∀ᵉ (p : ℕ) (hp_prime : p.Prime) (hp_nontrivial : p ≠ 2),
∃ q, q.Prime ∧ q < p ∧ orderOf (q : ZMod p) = p - 1formal-conjectures/985.lean ↗oeis
A002233 — a(1) = 1; for n > 1, a(n) = least positive prime primitive root of n-th prime.1,2,2,3,2,2,3,2,5,2,3,2,7,3,5,2,2,2,2,7,5,3,2,3,5,2,5,2,11,3,3,2,3,2,2,7,5,2,5,2,2,2,19,5,2,3,2,3,2,7,3,7,7,11,3,5,2,43,A103309 — Smallest prime primitive root of n that is less than n, or 0 if none exists.0,0,0,2,3,2,5,3,0,2,3,2,0,2,3,0,0,3,5,2,0,0,7,5,0,2,7,2,0,2,0,3,0,0,3,0,0,2,3,0,0,7,0,3,0,0,5,5,0,3,3,0,0,2,5,0,0,0,3,2,A219429 — Highest prime primitive root (less than p) for the n-th prime p. (or 0 if none exists).0,2,3,5,7,11,11,13,19,19,17,19,29,29,43,41,47,59,61,67,59,59,79,83,83,89,101,103,103,107,109,127,131,109,139,109,151,149
status
open