Problem 1107

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claims/: The 1 claim page of Problem 1107, one per claimant's result; the problem's standing derives from them.

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Statement. Let $r\geq 2$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$. Is every large integer the sum of at most $r+1$ many $r$-powerful numbers?

Status. Open.

Source. erdosproblems.com/1107, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1107, https://www.erdosproblems.com/1107.

References.

• [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.

Formalization. Statement in formal-conjectures.

Current assessment

The problem is open for every $r\geq3$. The case $r=2$ is Heath-Brown's theorem that every large integer is a sum of at most three 2-powerful numbers, an accepted partial claim on its claim page, credited by the site's curator on Problem 941.

Progress

At $r=2$ the question coincides with Problem 941, answered by Heath-Brown's theorem.