Problem 939

Let $r\geq 2$ . An $r$ -powerful number $n$ is one such that if $p\mid n$ then $p^r\mid n$ .

If $r\geq 4$ then can the sum of $r-2$ coprime $r$ -powerful numbers ever be itself $r$ -powerful? Are there at most finitely many such solutions?

Are there infinitely many triples of coprime $3$ -powerful numbers $a,b,c$ such that $a+b=c$ ?

Source: erdosproblems.com/939

Number theory
Powerful numbers

Status

Open

No claim settles this problem.

Open; the site's label is OPEN (page last edited 2026-05-28; proof-claims thread empty as of 2026-09-27). The site's remarks answer the third question yes through Nitaj [Ni95], with further constructions by Cohn [Co98] and Walsh [Wa24]; record solutions of the first question at $r=5$ (Cambie; Kitamura) and at $r=7$ and $r=8$ (Cambie); and record a construction (GPT-5.5 Pro, prompted by Price, 24 May 2026) giving infinitely many solutions for every $r\geq 6$ , so that the second question's answer is no there. Neither of the first two questions is answered at $r=4$ .

Co98: Cohn, J. H. E., A conjecture of Erdős on $3$ -powerful numbers. Math. Comp. 67 (1998), no. 221, 439-440. DOI: 10.1090/S0025-5718-98-00881-3.
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.
LaPa67: Lander, L. J. and Parkin, T. R., A counterexample to Euler's sum of powers conjecture. Math. Comp. 21 (1967), no. 97, 101-103. DOI: 10.1090/S0025-5718-1967-0220669-3.
Ni95: Nitaj, Abderrahmane, On a conjecture of Erdős on $3$ -powerful numbers. Bull. London Math. Soc. 27 (1995), no. 4, 317-318. DOI: 10.1112/blms/27.4.317.
Wa24: P. G. Walsh, A question of Erdős on 3-powerful numbers and an elliptic curve analogue of the Ankeny-Artin-Chowla conjecture. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 29 (2024), 83–87. DOI: 10.21857/y7v64t4jky. arXiv:2404.03970.

Problem 939

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Proof claims

Discussion