Problem 325

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claims/: The 1 claim page of Problem 325, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $\gg_\epsilon x^{3/k-\epsilon}$?

Status. 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); the paper does not cover $3\le k\le32$.

Source. erdosproblems.com/325, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #325, https://www.erdosproblems.com/325.

References.

• [ErMa38] Erdős, Pál and Mahler, Kurt, On the number of integers which can be represented by a binary form. Doc. Math. (2019), 475-481.
• [Wo15] Wooley, Trevor D., Sums of three cubes, II. Acta Arith. (2015), 73-100.

Formalization. Statement in formal-conjectures.

Current assessment

The standing is derived from the claim page in claims/: the problem is open, with one accepted partial claim. Browning and Heath-Brown's Corollary (Invent. Math. 2004), recorded on its claim page, gives for every $k\ge33$ asymptotically $\tfrac16cx^{3/k}$ integers $n\le x$ that are sums of three $k$th powers, with $c=\Gamma(1+1/k)^3/\Gamma(1+3/k)$, so $f_{k,3}(x)\gg x^{3/k}$ and both forms of the question hold in that range; the corollary rests on their count of $o(B^3)$ non-trivial solutions of $x_1^k+x_2^k+x_3^k=x_4^k+x_5^k+x_6^k$ with $\max x_i\le B$ for $k\ge33$. For $k=3$ the site's record is Wooley's Theorem 1.1 [Wo15], $f_{3,3}(x)\gg x^{0.91709477}$ for sums of three positive cubes (source card), short of the exponent $1$ the question asks for. The two-power analogue is Mahler and Erdős's theorem [ErMa38], which the site records as $f_{k,2}(x)\gg x^{2/k}$ for every $k\ge3$: an integral binary form of degree $k\ge3$ with nonzero discriminant represents $\gg u^{2/k}$ integers up to $u$ (source card); since zero is allowed as a summand, $f_{k,3}(x)\ge f_{k,2}(x)$. For $4\le k\le32$ no result beyond $f_{k,3}(x)\gg x^{2/k}$ is recorded, and later work that may extend Browning and Heath-Brown's range below $33$ is not assessed.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_2019_number_integers_represented_binary_form
• erdos_2019_number_integers_represented_binary_form / main_theorem
• erdos_2019_number_integers_represented_binary_form / section_4
• erdos_2019_number_integers_represented_binary_form / theorem_1
• wang_2021_sums_cubes_ratios_conjectures
• wang_2021_sums_cubes_ratios_conjectures / theorem_1_3
• wooley_2015_sums_three_cubes_ii
• wooley_2015_sums_three_cubes_ii / theorem_1_1
• wooley_2015_sums_three_cubes_ii / theorem_1_2