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Claim. Theorem 1.1 of J.-H. Fang and Y.-G. Chen, A quantitative form of the Erdős–Birch theorem, Acta Arith. 178 (2017), no. 4, 301--311, published online 10 May 2017, the date this page carries: for any coprime integers $p,q>1$ there are positive integers $K$ and $B$ with

$$\log_2\log_2K<q^{2p},\qquad \log_2\log_2\log_2B<q^{2p},$$

such that every integer $n\geq B$ is a sum of distinct terms of $\{p^aq^b: a\geq0,\ 0\leq b\leq K,\ a+b>0\}$. The statement and the surrounding history are recorded on the library's card.

Covers. The set in Theorem 1.1 is a subset of $\{p^aq^b\}$, so the theorem proves the statement of Problem 246, in its corrected Statement, which takes $a,b\geq2$, in a stronger form with an explicit threshold $B$.

Depends on. No page of this wiki.

Acceptance. Refereed: Acta Arithmetica, volume 178. Reviewed: the site's curator, Thomas Bloom, marks the problem PROVED and the commentary cites the paper among the quantitative forms (problem page last edited 7 December 2025); the curator had no part in the result. The problem's settling result is Birch's theorem. The proof was read but not independently verified.