Problem 325

Let $k\geq 3$ and $f_{k,3}(x)$ denote the number of integers $\leq x$ which are the sum of three nonnegative $k$ th powers. Is it true that

f_{k,3}(x) \gg x^{3/k}

or even $\gg_\epsilon x^{3/k-\epsilon}$ ?

Source: erdosproblems.com/325

Number theory
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Open

No claim settles this problem.

Open: the site's label (OPEN). Comments on the site's discussion thread of 2026-03-09 cite Browning and Heath-Brown, whose corollary settles every $k\ge33$ (claim page (Browning Heath Brown, 2004)); the paper does not cover $3\le k\le32$ .

Problem 325

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