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Updated Oct 7

Problem 322

claims/: The 1 claim page of Problem 322, one per claimant's result; the problem's standing derives from them.

Statement. Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$ th powers. What is the order of growth of $1_A^{(k)}(n)$ , i.e. the number of representations of $n$ as the sum of $k$ many $k$ th powers? Does there exist some $c>0$ and infinitely many $n$ such that

1_A^{(k)}(n) >n^c?

Status. Open.

Source. erdosproblems.com/322, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #322, https://www.erdosproblems.com/322.

References.

[Er36] Erdős, Paul, On the Representation of an Integer as the Sum of k k-th Powers. J. London Math. Soc. (1936), 133-136.
[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
[Gu04] Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section D4 "Waring's problem. Sums of $l$ $k$ th Powers.", printed p. 229: Mahler's disproof of Hypothesis K for $k=3$ , "Erdős thinks it possible that for all $n$ , $r_{3,3}<c_2n^{1/12}$ but nothing is known", and the Chowla--Erdős bound $r_{k,k}>\exp(c_k\ln n/\ln\ln n)$ for infinitely many $n$ ; the page's question is not stated there. Library home: guy_2004_unsolved_problems_number_theory.
[Ma36] Mahler, Kurt, Note on Hypothesis K of Hardy and Littlewood. J. London Math. Soc. (1936), 136-138.

Formalization. Statement in formal-conjectures.

Current assessment

For $k=3$ the second question has a refereed positive answer: Mahler [Ma36] shows that every large twelfth power $N$ is a sum of three cubes in at least $9^{-1/3}N^{1/12}$ ways, so $1_A^{(3)}(n)>n^c$ holds for infinitely many $n$ for every $c<1/12$ (its claim page). The order of growth for $k=3$ and both questions for $k\ge4$ are open; the site's commentary records Erdős's belief that Hypothesis K fails for every $k\ge4$ . The bound $1_A^{(k)}(n)\gg n^{c/\log\log n}$ for infinitely many $n$ , proved independently by Erdős [Er36] and Chowla for every $k\ge3$ , is smaller than every fixed power of $n$ , so it settles no instance of the second question and has no claim page.

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.

Linked from (13)

Mahler's disproof of Hypothesis K for cubes problems/diophantine_problems/E0322/claims Diophantine Problems and Powers Diophantine Problems and Powers diophantine_problems/erdos_1936_representation_integer_as_sum_th_powers Inequality (1) (p. 133): infinitely many m with f(m) > exp(c_1 log m / log log m)Section 4 (p. 136): the same lower bound for a_1 x_1^(k_1) + ... + a_l x_l^(k_l) when the 1/k_i sum to 1 diophantine_problems/mahler_1936_note_hypothesis_k_hardy_littlewood Equation (2): a one-parameter family of solutions of x^3 + y^3 + z^3 = 1 Theorem (p. 138, unnumbered): Hypothesis K is false for cubes Theorem (p. 138, unnumbered): many solutions of a signed diagonal equation of degree n number_theory/erdos_1965_recent_advances_current_problems_number_theory number_theory/guy_2004_unsolved_problems_number_theory

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