Problem 940

Statement

Let $r\geq 3$ . A number $n$ is $r$ -powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$ .

Are there infinitely many integers which are not the sum of at most $r$ many $r$ -powerful numbers? Does the set of integers which are the sum of at most $r$ $r$ -powerful numbers have density $0$ ?

Source: erdosproblems.com/940

Number theory
Powerful numbers

Status

Open

No claim settles this problem.

Open. The infinitude question is answered for $r=3$ and $r=4$ by accepted partial claims, in both cases with positive lower density of the complement; the higher exponents and the density-zero question remain open. The erdosproblems.com label was OPEN, and the page was last edited on 2025-11-03. The erdosproblems.com remarks record that Baker and Brüdern [BaBr94] first proved the density-zero statement at $r=2$ . They also record that at $r=3$ the statement is unknown even for sums of three cubes. Finally, they record that Heath-Brown [He88] proves every large integer is a sum of at most three $2$ -powerful numbers (Problem 941). On 2026-10-06 the erdosproblems.com proof-claims tab carried one partial claim, filed 2026-09-06.

BaBr94: Baker, R. C. and Brüdern, J., On sums of two squarefull numbers. Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 1, 1-5.
Er76d: Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.
He88: Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. Séminaire de Théorie des Nombres, Paris 1986–87, Progr. Math. 75, Birkhäuser Boston (1988), 137-163.

Problem 940

Statement

Status

Proof claims

Discussion