Problem 980

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claims/: The 2 claim pages of Problem 980, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $c_k>0$?

Statement (precise). Let $k\geq 2$ and, for primes $p\equiv1\pmod k$, let $n_k(p)$ denote the least $k$th power nonresidue of $p$; set $n_k(p)=0$ for every other prime. Is it true that

$$\sum_{p<x} n_k(p)\sim c_k \frac{x}{\log x}$$

for some constant $c_k>0$?

Notes. The site's wording defines $n_k(p)$ as the least $k$th power nonresidue of $p$ for every prime $p$, as Erdős's texts do ((79) of [Er65b], the site's source, printed p. 232; conjecture (4) of [Er61e], p. 11), and leaves it undefined at the primes with $\gcd(k,p-1)=1$, which have no $k$th power nonresidue. Read as a sum over the primes that have one, it includes, for composite $k$, the primes $p\not\equiv1\pmod k$ with $d=\gcd(k,p-1)>1$, whose least $k$th power nonresidue is $n_d(p)$; for prime $k$ only the primes $p\equiv1\pmod k$ contribute. Elliott [El67b] defines $n_k(p)$ in the paper's introduction for $p\equiv1\pmod k$ and sets $n_k(p)=0$ for every other prime, and the paper's Theorem 1 proves the asymptotic under that convention for every $k\ge2$ (indeed with any exponent $a<4e^{1-1/k}$ in place of $1$, and with the explicit constant $\sum_r k^{-r}q_r$ over the primes $q_r$ when $k$ is an odd prime). The site labels the problem PROVED and its commentary says "The general case was proved by Elliott [El67b]", without remarking on the convention. The curator therefore reads the sum as Elliott does, and the precise Statement adopts Elliott's convention. The change inserts Elliott's definition of $n_k(p)$; nothing else changes. For prime $k$ the two readings agree, since a prime $p\not\equiv1\pmod k$ then has every residue a $k$th power. For composite $k$ they differ: under the precise Statement the problem is proved for every $k\ge2$ by Elliott's Theorem 1 (claim page); under the site's wording the contribution of the primes $p\not\equiv1\pmod k$ with $\gcd(k,p-1)>1$ is covered by no recorded source, so that reading is open for composite $k$ and is recorded as a variant under Formulation. Erdős [Er61e] proved the case $k=2$, where the readings agree, with $c_2=\sum_j p_j/2^j$ (claim page). The curator's reading is inferred from the label and the credit alone; no text of the curator's states the convention.

Formulation. The site's wording, read as a sum of the least $k$th power nonresidue over every prime that has one, is a variant of the precise Statement. For prime $k$ the two coincide. For composite $k$ the variant adds the primes $p\not\equiv1\pmod k$ with $d=\gcd(k,p-1)>1$, each contributing $n_d(p)$. Theorem 1 of [El67b] does not cover that sum, and no recorded source does, so the variant is open for composite $k$. It has no claim page and does not enter the standing.

Status. PROVED on erdosproblems.com, crediting Elliott [El67b], who proved the asymptotic for every $k$ under the convention of the precise Statement, with a constant $C_{k,1}$ given as an explicit prime series when $k$ is an odd prime (claim page), after Erdős [Er61e] had proved the case $k=2$ (claim page) and conjectured the general one. The label describes the precise Statement; the variant under Formulation stays open for composite $k$.

Source. erdosproblems.com/980, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #980, https://www.erdosproblems.com/980.

References.

• [El67b] Elliott, P. D. T. A., A problem of Erdős concerning power residue sums. Acta Arith. 13 (1967), 131-149.
• [Er61e] Erdős, Pál, Remarks on number theory. I. Mat. Lapok (1961), 10-17.
• [Er65b] Erdős, P., Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.

Formalization. None recorded.

Current assessment

Scope. The standing rests on Elliott's refereed paper, Erdős's refereed case $k=2$ and the site's credit, as the claim pages record; the corpus has not checked the proofs, and no status search beyond the site is recorded. The target of the standing is the precise Statement. Elliott's theorem is an accepted full claim on it and Erdős's case $k=2$ an accepted partial one, so the derived standing is proved. The variant for composite $k$ (see Formulation) stays open and does not enter the standing.

A release manuscript on least nonresidues. The OpenAI mathematics release's manuscript Deterministic Polynomial Factorization over Prime Fields (4 October 2026; folder preprints/Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026 of github.com/openai/math, pinned by that link; library card openai_2026_deterministic_polynomial_factorization_over_prime_fields) derives in its Proposition 11.3 (Section 11), conditionally on the zero-free strip for Hecke $L$-functions claimed in the release's companion manuscript on primitive roots, bounds on the size of a prime $\ell$ modulo which a given large prime $p$ is not a $q$th power. It concerns individual primes, not the averaged asymptotic this problem asks for, and it states no result on this problem; it is background here, nothing in it is verified in this corpus, and it has no claim page.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• elliott_nd_problem_erdos_concerning_power_residue_sums
• erdos_1961_szamelmeleti_megjegyzesek
• erdos_1961_szamelmeleti_megjegyzesek / conjecture_4
• erdos_1961_szamelmeleti_megjegyzesek / equation_3
• openai_2026_deterministic_polynomial_factorization_over_prime_fields
• openai_2026_deterministic_polynomial_factorization_over_prime_fields / proposition_11_3