The board / number theory / #980
Erdős Problem #980
PROVEDdatabase updated 2025-09-03
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
\(n_k(p)\): the least positive integer which is not a \(k\)-th power residue modulo \(p\). If every integer is a \(k\)-th power residue (for example if \(\gcd(k, p - 1) = 1\)) the value is \(0\); this does not happen for the primes \(p \equiv 1 \pmod k\) with \(k \geq 2\) used below.
\(\sum n_k(p)\) over the primes \(p < x\) with \(p \equiv 1 \pmod k\).
Let \(k\geq 2\) and \(n_k(p)\) denote the least \(k\)th power nonresidue of \(p\). Is it true that \[\sum_{p<x} n_k(p)\sim c_k \frac{x}{\log x}\] for some constant \(c_k>0\)? Erdős [Er65b] asked this; it is true, and was proved by Elliott [El67b]. As in [El67b], the sum is over the primes \(p \equiv 1 \pmod k\) (for each of which a \(k\)th power nonresidue exists).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/980.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/980
Attempts
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