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Claim. Let $a,b>1$ be coprime integers. Every sufficiently large integer is a sum of distinct integers of the form $a^kb^l$ with $k,l\geq 0$, the statement of Problem 246, in its corrected Statement, which takes $a,b\geq2$. Cassels obtains it from his Theorem I: if a set $A$ of positive integers satisfies $(A(2n)-A(n))/\log\log n\to\infty$, where $A(n)$ counts the elements of $A$ up to $n$, and $\sum_{\alpha\in A}\lVert\alpha\theta\rVert^2$ diverges for every real $\theta\in(0,1)$, then every sufficiently large integer is a sum of distinct elements of $A$. The paper's introduction (p. 111) says that the investigation was touched off by Birch's paper on the sums of the numbers $p^\alpha q^\beta$ for coprime $p,q$, and that Birch's results are an immediate consequence of Theorem I; the verification of the two hypotheses for the set $\{a^kb^l\}$ is Cassels's and was not reconstructed here. The paper's digest is the library card, which records [[../library/integer_sequences/cassels_1960_representation_integers_as_sums_distinct_summands/theorem_i|Theorem I]] and the passage on Birch as read clause by clause on the printed pages, and the proof of Theorem I (Section 2, the Hardy--Littlewood circle method applied to a thinned subset of $A$) as read for pointers but not checked step by step.

Source. J. W. S. Cassels, On the representation of integers as the sums of distinct summands taken from a fixed set, Acta Sci. Math. (Szeged) 21 (1960), 111--124, received 3 September 1959. The volume carries the year and no day, so this page is dated by the first day of 1960; the result follows Birch's, whose page is Birch's theorem, and the two are recorded apart because Cassels proves the statement by his own, more general criterion rather than by Birch's argument.

Acceptance. Refereed: the result is a journal paper in Acta Scientiarum Mathematicarum. Reviewed: the site's curator, Thomas Bloom, marks the problem PROVED and records that it also follows from Cassels's later, more general result (problem page last edited 7 December 2025); the curator had no part in the result. Nothing here is this project's own review.