Sums of powerful numbers

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source_notes/: Paper summaries and source comparisons used in the research on Problem 940.

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Where things stand

Problem 940 asks, for $r\ge3$, whether infinitely many integers are not sums of at most $r$ $r$-powerful numbers, and whether those sums have density zero. The site lists it as open. For $r=3$, an unpublished manuscript of Beyer de Ryke claims positive lower density for the complement, and Wang shows, assuming three conjectures on Hasse--Weil $L$-functions, that sums of three cubes have positive lower density. No result for $r\ge4$ is recorded. The source notes summarize the published work used in reading the problem.