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Claim. For coprime integers $p,q>1$, the set $\{p^aq^b: a\geq0,\ 0\leq b\leq K\}$ is complete for an explicit $K$ with

$$K(p,q)\leq2p^{2d^{2^{2q^{4p+3}}}},\qquad d=1152\log_2p\log_2q,$$

where $K(p,q)$ is the least $K$ for which the set is complete. This is the result of N. Hegyvári, On the completeness of an exponential type sequence, Acta Math. Hungar. 86 (2000), no. 1--2, 127--135, as Fang and Chen record it in the introduction of their quantitative form (p. 301). The issue is dated January 2000 in its record, with no day, so this page carries the first of that month. The paper is not held, and the statement is recorded from that citation.

Covers. Since the set is a subset of $\{p^aq^b\}$, the theorem proves the statement of Problem 246, in its corrected Statement, which takes $a,b\geq2$, in a stronger form that bounds the exponent of the second base.

Depends on. No page of this wiki.

Acceptance. Refereed: Acta Mathematica Hungarica, volume 86. Reviewed: the site's curator, Thomas Bloom, marks the problem PROVED and his commentary credits Hegyvári with the first explicit bound (problem page last edited 7 December 2025); the curator had no part in the result. The problem's settling result is Birch's theorem.