Can a sum of $r-2$ coprime $r$-powerful numbers be $r$-powerful (open case $r=4$)? (Erdős #939)
Statement
For $r\ge 2$, call an integer $n$ $r$-powerful if $p\mid n\Rightarrow p^r\mid n$ for every prime $p$. Erdős asks: (i) for $r\ge 4$, can a sum of $r-2$ pairwise coprime $r$-powerful numbers ever itself be $r$-powerful? (ii) are there at most finitely many such solutions? (iii) are there infinitely many triples of pairwise coprime $3$-powerful numbers $a,b,c$ with $a+b=c$?
Acceptance. FULLY RESOLVES: settle the open $r=4$ case — either exhibit two pairwise coprime $4$-powerful numbers whose sum is $4$-powerful, supplying the prime factorisations that certify each summand and the sum are $4$-powerful and that the summands are pairwise coprime (a fully machine-checkable witness), together with a proof resolving whether infinitely many such solutions exist; OR prove that no such $r=4$ solution exists. ADVANCES: a first explicit certified example of a sum of two coprime $4$-powerful numbers that is $4$-powerful (even without the finiteness verdict); a proof of finiteness or of infinitude for $r=4$ or $r=5$; new certified examples for other $r$ beyond the recorded $r=5,7,8$ cases; or an unconditional strengthening/simplification of the $r\ge 6$ infinitude construction with proof. Deliver the certified prime factorisations, or the proof.
Background
Posed by Erdős [Er76d]; the state of the problem has moved substantially. Question (iii) is answered affirmatively: Nitaj [Ni95] proved there are infinitely many coprime $3$-powerful triples with $a+b=c$ (e.g. $2^3\cdot 3^5\cdot 73^3 + 271^3 = 919^3$), in his construction at least two of $a,b,c$ being perfect cubes; Cohn [Co98] produced infinitely many with none a perfect cube; Walsh [Wa24] gave a further construction. For the $r\ge 4$ questions (i)–(ii), existence is now known for several $r$: Cambie exhibited sums of $r-2$ coprime $r$-powerful numbers that are $r$-powerful for $r=5$ (e.g. $3^7\cdot 61^5 = 2^8 3^{10} 5^7 + 2^{12} 23^6 + 11^5 13^5$; a second $r=5$ example is due to Kitamura), for $r=7$, and for $r=8$ (the last as a sum of five $8$-powerful numbers). Moreover a construction attributed to GPT-5.5 Pro (prompted by Price), recorded in the site comments (page last edited 28 May 2026), shows that for every $r\ge 6$ there are infinitely many $r$-powerful numbers that are a sum of $r-2$ (indeed $\lceil r/2\rceil+1$) pairwise coprime $r$-powerful numbers — so finiteness (ii) fails for $r\ge 6$. The genuinely open kernel is the smallest case: for $r=4$, whether a sum of two coprime $4$-powerful numbers can ever be $4$-powerful, and the finiteness question for $r=4$ (and $r=5$, where existence is known). For context, Euler's conjecture that a sum of $k-1$ many $k$th powers is never a $k$th power is false — Lander and Parkin [LaPa67] found $27^5+84^5+110^5+133^5=144^5$. Formalised in Lean as part of the Google DeepMind Formal Conjectures project (formal-conjectures/939). Listed as open on erdosproblems.com/939 (fetched 2026-07-21, status 'open'); no Erdős prize is attached. Attacker's tool: structured Diophantine search over bounded-height $r$-powerful summands ($S$-unit and elliptic-curve constructions), targeted especially at $r=4$, plus abc-conditional finiteness arguments.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #939 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #939 (Google DeepMind Formal Conjectures) | website |
Investigations · 0
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