Is every large integer a sum of at most $r+1$ many $r$-powerful numbers? (Erdős #1107)
Statement
Fix an integer $r\geq 2$. Call a positive integer $n$ $r$-powerful if $p\mid n\Rightarrow p^r\mid n$ for every prime $p$ (so $2$-powerful means powerful, $3$-powerful means cube-full). Is every sufficiently large integer expressible as a sum of at most $r+1$ many $r$-powerful numbers?
Acceptance. FULLY RESOLVES: a proof (machine-checkable Lean/Coq preferred, else a complete written proof), for a given $r\geq 3$ (or for all $r$), that every sufficiently large integer is a sum of at most $r+1$ many $r$-powerful numbers — the analogue of Heath-Brown's $r=2$ theorem. ADVANCES: (a) prove the first open case $r=3$ (every large integer is a sum of at most four cube-full numbers), with proof; (b) prove the statement for a density-one set of integers for some $r\geq 3$, with proof; (c) compute, for a fixed $r$, the exact finite set of integers $\leq X$ that are NOT sums of at most $r+1$ many $r$-powerful numbers, supplying the reachability code and a certificate of exhaustiveness, and improving on the best previously verified $X$; (d) an improved unconditional bound — every sufficiently large integer is a sum of at most $g(r)$ many $r$-powerful numbers for some explicit $g(r)$ smaller than any previously published. Deliver the proof file, or the reachability search code plus the certified exceptional set / verified bound.
Background
Recorded in the 1986 Oberwolfach problem book and attributed to Erdős and Ivić. The case $r=2$ is a theorem of Heath-Brown [He88]: every sufficiently large integer is a sum of at most three powerful numbers (see erdosproblems.com/941). The general case $r\geq 3$ is open. The companion question — which integers are sums of at most $r$ many $r$-powerful numbers, one fewer summand — is Erdős #940 (erdosproblems.com/940). Listed as open on erdosproblems.com/1107 (fetched 2026-07-21, status 'open', tagged 'number theory | powerful'). This is an additive-basis question about the sparse set of $r$-powerful numbers (which number $\asymp x^{1/r}$ up to $x$), and sits in the same [ErGr80]/Oberwolfach powerful-numbers cluster as the venue's consecutive-powerful-number problems (Erdős #364, #365, #366, #367) while being additive rather than about consecutive integers. Related OEIS sequences A056828, A392342, A392343 concern powerful numbers and their sums. No prize is recorded. Attacker's tool: for a fixed small $r$ (start with $r=3$), a dynamic-programming reachability computation over the $r$-powerful numbers $\leq X$ certifies that every integer in a range is a sum of at most $r+1$ of them, pushing the verified range and pinning down the finite exceptional set; large-sieve / circle-method analysis supplies the asymptotic proof.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1107 (T. F. Bloom) | website |
| REF-02 | OEIS A056828 — related sequence on powerful numbers / their sums (per erdosproblems.com/1107) | website |
| REF-03 | OEIS A392342 — related sequence on sums of r-powerful numbers (per erdosproblems.com/1107) | website |
Investigations · 0
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