Problem 939

../

claims/: The 4 claim pages of Problem 939, one per claimant's result; the problem's standing derives from them.

---

Statement. 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 $a+b=c$?

Formulation. "Coprime" is read as joint coprimality: the summands have greatest common divisor $1$, not pairwise. The first two questions are read for each $r\geq4$, so each is answered only when it is decided for every $r\geq4$. Erdős's source states the conjecture for each $r$ with no coprimality condition, "the sum of $r - 2$ $r$-powerful numbers is never (or at most finitely often) $r$-powerful" [Er76d, p. 33], and writes the three-term case with $(u_i,u_j,u_\ell)=1$, where joint and pairwise coprimality coincide. The site adopted Cambie's and Kitamura's $r=5$ examples, which are jointly but not pairwise coprime, and formal-conjectures uses joint coprimality (Finset.Coprime) for every $r\geq4$. Under a pairwise reading, the $r=5$ examples and the $r\geq6$ construction would not count. Under a "for some $r$" reading, the first question would already be answered yes at $r=5$.

Status. 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 $r=4$.

Source. erdosproblems.com/939, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #939, https://www.erdosproblems.com/939.

References.

• [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.

Formalization. Statement in formal-conjectures FormalConjectures/ErdosProblems/939.lean (linked at the commit of 2026-09-18, current on 2026-09-27): erdos_939 (research open) states the first question as answer(sorry) ↔ ∀ r ≥ 4, (Erdos939Sums r).Nonempty; since 2026-09-09 Erdos939Sums requires positive, distinct, jointly coprime summands; erdos_939.variants.finite is marked answer(False) and erdos_939.variants.infinite_of_six_le (∀ r ≥ 6, (Erdos939Sums r).Infinite) is marked research solved with proof sorry (since 2026-09-11); erdos_939.variants.triples is answer(True). Conjectures.io record 91915fc3-9040-4a31-8da5-b49d4e2cc2fb (certified 6 Aug 2026, partial award for a formalization defect): its Lean proof establishes the pre-positivity statement by taking $\{0,1\}$ at $r=4$; the site's review states it does not settle the problem; the file's infinite_rpowerful_sums theorem is a kernel-checked proof of the $r\geq 6$ infinitude with positive summands. See the Conjectures.io card.

Current assessment

Site formulation as of 2026-09-27 (last edited 28 May 2026); three parts as quoted in the Statement, read as the Formulation records. The page-level status is open because $r=4$ is undecided for both of the first two parts.

Part 3 is resolved (yes) by the refereed sources Nitaj 1995 and Cohn 1998; Walsh 2024 (Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 29, 2024) gives a further construction.

A manuscript of the OpenAI Math Release, The Selmer converse for elliptic curves at every prime (OpenAI, 2026-09-24; published in the release at https://github.com/openai/math/blob/adc7f1241/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026/main.pdf, with its intake card at openai_2026_selmer_converse_elliptic_curves_at_prime), states as an application at the prime $3$ (its Corollary 10.1) that for every prime $\ell\equiv4,7,8\pmod 9$ the curve $X^3+Y^3=\ell Z^3$ has Mordell–Weil rank one. That curve has the Weierstrass model $Y^2=X^3-432\ell^2$, whose positive rank is the hypothesis of Walsh's Theorem 1.1, so for those primes Walsh's construction gives infinitely many pairwise coprime solutions of $x^3+y^3=\ell^4z^3$, hence coprime $3$-powerful triples; the Walsh card records Logan's Selmer computation pointing to rank one in exactly these classes. The release names no Erdős problem and claims nothing about this one; the observation bears only on part 3, which the refereed sources settle, and not on the open cases $r=4$ and $r=5$. It is recorded as context at the level of the release's own statement, with no verification recorded; the release has no Lean for it and claims nothing about this problem, so no claim page is written for it.

Part 1 (can it ever happen for $r\geq 4$): instances exist at $r=5$ (Cambie; Kitamura), at $r=7$ and $r=8$ (Cambie, asserted in the site text with the numbers not displayed), and for every $r\geq 6$ (the construction below); none is known at $r=4$, where the only evidence is a forum-reported exhaustive search bound $\max(a,b)>10^{14}$.

Part 2 (at most finitely many?): false for every $r\geq 6$ (infinitely many solutions); open at $r=4$ and $r=5$.

Acceptance evidence. Part 3 rests on two refereed journal papers (Nitaj, Cohn); Walsh's refereed construction is conditional on a positive-rank hypothesis. The $r\geq 6$ construction is a forum post adopted into the site text and formalized in Lean (Price 2026); the Lean text was kernel-checked as part of the Conjectures.io submission 91915fc3, whose accepted target was a defective formal statement (no positivity) and whose review explicitly declines to treat it as a resolution; no refereed publication of the $r\geq 6$ result was found. The $r=4$ case has no source of any kind.

Claim pages. Nitaj's and Cohn's answers to part 3 are accepted partial claims, refereed, on Nitaj's claim page and Cohn's claim page. Walsh's Theorem 1.1 is an accepted conditional claim on its claim page: it applies to each odd prime $p$ at which $Y^2=X^3-432p^2$ has positive rank, and no refereed source cited here exhibits such a prime (the release manuscript above states rank one for $p\equiv4,7,8\pmod 9$, with no verification recorded). The $r\geq 6$ construction is a pending partial claim on Price's claim page; it settles neither of the first two parts, both of which stay open at $r=4$ and $r=5$. CrowdMath's Theorem 2.1 (abc implies only finitely many coprime $4$-powerful $a+b=c$) has no claim page: it rests on the unproved abc conjecture, reaches only $r=4$ and says nothing at $r=5$, so it settles no part even conditionally. Cambie's and Kitamura's examples and Cambie's $r=4$ search have no claim pages: they come from the site's thread and remarks, with no manuscript.

Dated search scope, 2026-09-27: erdosproblems.com (problem page, revision history, forum thread 939 with 9 comments, proof-claims thread empty); the community database (open, last update 2025-08-31); conjectures.io (result, solution, problem page, public verification-report API, and the validator repository's review-decision document); formal-conjectures (main and the commit history of 939.lean); arXiv (2404.03970 abstract; API search for "powerful numbers" with Erdős, four hits, none on this problem); Crossref (the three DOIs). X not searched.

Local checks: the four displayed identities (Cambie $r=5$, Kitamura $r=5$, Nitaj, Lander–Parkin) were recomputed, including powerfulness and gcds; the Conjectures.io Lean file's SHA-256 matches the page's printed digest, and this corpus has not built it; the manuscript's one-page proof of the $r\geq 6$ construction is followed step by step on the Price card.

The manuscript's proof is written out in the corpus's own words, with the distinctness step it omits supplied and labeled, in the research folder for Problem 939; that reconstruction is author-recorded and changes no status.

Known results

• Third question (infinitely many coprime $3$-powerful $a,b,c$ with $a+b=c$): yes. Nitaj (Bull. London Math. Soc. 27 (1995), 317–318, refereed) constructs infinitely many, e.g. $2^3\cdot3^5\cdot73^3+271^3=919^3$, with at least two of $a,b,c$ perfect cubes; Cohn (Math. Comp. 67 (1998), 439–440, refereed) constructs infinitely many with none a perfect cube; Walsh (Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 29 (2024), 83–87; arXiv:2404.03970) gives an elliptic-curve construction. This part is resolved by refereed literature.
• First question ($r\geq 4$: can a sum of $r-2$ coprime $r$-powerful numbers be $r$-powerful?): yes for $r=5,7,8$ by explicit examples recorded on erdosproblems.com. At $r=5$, Cambie's $3^7\cdot61^5=2^8\cdot3^{10}\cdot5^7+2^{12}\cdot23^6+11^5\cdot13^5$ (joint gcd $1$; the first two summands share $2^8$) and Kitamura's $3^7\cdot13^7=2^5\cdot17^6+7^{11}+2^5\cdot3^6\cdot7^8$ (forum post 6641, 25 May 2026, with a linked verification repository https://github.com/KitaKen1/erdos-939); both identities were recomputed, and Cambie's is checked in Lean in formal-conjectures erdos_939.variants.examples and in the Conjectures.io file's leg_five. Yes for every $r\geq 6$ by the construction below. Open at $r=4$: no example is known; Cambie reports (forum post 6654, 26 May 2026, no code linked, unverified) an exhaustive search showing that any coprime $4$-powerful $a+b=c$ has $\max(a,b)>10^{14}$.
• Second question (at most finitely many such solutions?): no for every $r\geq 6$. The construction (GPT-5.5 Pro prompted by Price, forum post 6640, 24 May 2026, adopted into the site text on 28 May 2026) expands $(X+Y)^r=(X-Y)^r+\sum_{j\ \mathrm{odd}}2\binom rjX^{r-j}Y^j$, splits the $j=3$ term into $t=\lfloor r/2\rfloor-2\geq 1$ distinct positive multiples of $X^{r-3}Y^3$ to reach exactly $r-2$ positive summands, and takes $X=q^r$, $Y=B^r$ with $B$ divisible by every prime in the coefficients and $q>B$ a prime not dividing $B$: every summand and the sum are $r$-powerful, $\gcd(X-Y,XY)=1$ gives joint coprimality, and varying $q$ gives infinitely many. It is kernel-checked in Lean (Mathlib) as infinite_rpowerful_sums (positive, $r$-powerful, distinct, jointly coprime summands; infinitely many $r$-powerful sums) inside the Conjectures.io-verified file, whose lines 1–676 coincide with the autoformalization linked from the forum post; formal-conjectures records it as erdos_939.variants.infinite_of_six_le (research solved, proof sorry) and marks erdos_939.variants.finite answer(False). Finiteness remains open at $r=4$ (no example) and $r=5$ (two examples known; Cambie's search, forum post 6654, finds no further solution with $d<10^{16}$ except possibly ones with all pairwise gcds $>1$).
• Conjectures.io record 91915fc3 (6 Aug 2026): a Lean proof of the formal-conjectures statement ∀ r ≥ 4, (Erdos939Sums r).Nonempty in the version that omitted positivity of the summands; the $r=4$ leg is the degenerate set $\{0,1\}$. The site's review classed it a formalization-defect award (partial) and states that it does not settle Problem 939; the task was withdrawn and the catalog added positivity on 2026-09-09. It is not acceptance of the catalog question.
• Context: Euler's conjecture that a sum of $k-1$ $k$-th powers is never a $k$-th power fails at $k=5$ [LaPa67]: $27^5+84^5+110^5+133^5=144^5$ (recomputed). Cambie (forum post 1047, 13 Oct 2025) notes that the $n$-conjecture in its known forms is too weak to force finiteness in the second question.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• cohn_1998_conjecture_erdos_3_powerful_numbers
• conjectures_io_2026_erdos_939_lean_r_powerful_sums
• conjectures_io_2026_erdos_939_lean_r_powerful_sums / conjectures_io_2026_erdos_939_lean_r_powerful_sums
• corvaja_zannier_2011_abcd_function_fields
• corvaja_zannier_2011_abcd_function_fields / recalled_abc_abcd_bounds
• nitaj_1995_conjecture_erdos_3_powerful_numbers
• openai_2026_selmer_converse_elliptic_curves_at_prime
• openai_2026_selmer_converse_elliptic_curves_at_prime / corollary_10_1
• openai_2026_selmer_converse_elliptic_curves_at_prime / theorem_1_1
• price_2026_infinite_r_powerful_sums
• price_2026_infinite_r_powerful_sums / theorem
• walsh_2024_question_erdos_powerful_numbers_elliptic_curve