Integers that are no sum of $r$ many $r$-powerful numbers: infinitely many, sumset density 0? (Erdős #940)
Statement
Let $r\ge 3$ and call $n$ $r$-powerful if $p\mid n\Rightarrow p^r\mid n$ for every prime $p$. Two questions: (i) are there infinitely many positive integers that are not expressible as a sum of at most $r$ many $r$-powerful numbers? (ii) does the set of integers that are a sum of at most $r$ many $r$-powerful numbers have natural density $0$?
Acceptance. FULLY RESOLVES: complete proofs — machine-checkable preferred, else full written proofs — settling both (i) and (ii) for all $r\ge 3$; OR a disproof of either. ADVANCES: prove (i) for some fixed $r\ge 3$ (infinitely many integers not a sum of at most $r$ many $r$-powerful numbers); prove the density-$0$ statement (ii) for some fixed $r\ge 3$ (e.g. $r=3$); establish an upper bound on the counting function of the representable set below $x$ that is strictly stronger than what background records, with proof; or extend a reproducible computation of non-representable integers and density estimates to a new range with a certificate. Deliver the proof, or the bound/computation together with its certificate.
Background
Posed by Erdős [Er76d, p.33] and recorded in the Oberwolfach problem book in 1986 as a problem of Erdős and Ivić. For $r=2$ the density-$0$ claim is comparatively easy: it was first proved in the literature by Baker and Brüdern [BaBr94], and an elementary argument is given by Tao in the site comments. For $r=3$ it is not even known whether the integers expressible as a sum of at most three cubes have density $0$ — a well-known hard problem — so question (ii) is wide open for every $r\ge 3$. Erdős [Er76d] asserted that a 'simple counting argument' yields infinitely many integers that are not a sum of at most $r$ many $r$-powerful numbers, but Schinzel pointed out an error, leaving (i) open. In the opposite direction, Heath-Brown [He88] proved that every sufficiently large integer is a sum of at most three $2$-powerful numbers (Erdős #941, erdosproblems.com/941). See also Erdős #1081 (a refinement for $r=2$) and #1107 (the case of $r+1$ summands). Formalised in Lean as part of the Google DeepMind Formal Conjectures project (formal-conjectures/940). Listed as open on erdosproblems.com/940 (fetched 2026-07-21, status 'open'); no Erdős prize is attached. Attacker's tool: large-scale computation of the representation sets (enumerate $r$-powerful numbers, form their $\le r$-fold sumsets, measure counting densities and locate non-representable integers) to gather evidence and calibrate the density, alongside circle-method and sieve analysis.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #940 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #940 (Google DeepMind Formal Conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.