Problem 951

Let $1<a_1<\cdots$ be a sequence of real numbers such that

\left\lvert \prod_i a_i^{k_i}-\prod_j a_j^{\ell_j}\right\rvert \geq 1

for every distinct pair of non-negative finitely supported integer tuples $k_i,\ell_j\geq 0$ . Is it true that

\#\{ a_i \leq x\} \leq \pi(x)?

Source: erdosproblems.com/951

Number theory

Wiki page

Status

Open

No claim settles this problem.

Open. The site labels the problem OPEN (page last edited 06 April 2026) and says that no finite computation can resolve it; its commentary states that the wording leaves the quantifier over $x$ implicit, that the reading for every $x$ fails by a finite calculation, and credits the counterexample at $x=10$ to ChatGPT-5.2 Pro prompted by Leeham. The reading for every $x$ is answered in the negative by that counterexample, a pending partial claim which the curator's commentary credits; the further finite counterexamples of 28 January and 1 February 2026 ( $x=5-\varepsilon$ , $7-\varepsilon$ , $11-\varepsilon$ , $13-\varepsilon$ , $17-\varepsilon$ ) are a pending partial claim for the same part. The reading for all sufficiently large $x$ , the one the site's label and the formal-conjectures statement follow, is open: no proof, disproof, refereed partial result or proof claim for it was found in the search whose scope the Current assessment records, and nothing found bears on it beyond the elementary integer case. The derived standing is open.

Er69: Erdős, P., Some applications of graph theory to number theory. The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968), Lecture Notes in Mathematics 110, Springer (1969), 77--82. Printed p. 82: the closing question; printed p. 79: display (6), the integer subset-product bound. Library home: Erdős 1969; result page Erdős 1969, Inequality (6).
Er77c: Erdős, P., Problems and results on combinatorial number theory. III. Number Theory Day (Proc. Conf., Rockefeller Univ., New York, 1976), Lecture Notes in Mathematics 626, Springer (1977), 43--72. Printed pp. 68--69. Library home: Erdos 1977.
Er80: Erdős, P., A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115. Printed pp. 102--104. Library home: Erdos 1980.
Di77: Diamond, H. G., When do Beurling generalized integers have a density? J. Reine Angew. Math. 295 (1977), 22--39 (Crossref record, DOI 10.1515/crll.1977.295.22; [Er80] p. 104 prints 22--29); cited by [Er80] (p. 104) as the entry point to the Beurling literature. Not held; it gives background, not a result this page uses.

Problem 951

Status

Proof claims

Discussion