Problem 399

../

claims/: The 4 claim pages of Problem 399, one per claimant's result; the problem's standing derives from them.

---

Statement. Is it true that there are no solutions to

$$n! = x^k\pm y^k$$

with $x,y,n\in \mathbb{N}$, with $xy>1$ and $k>2$?

Status. DISPROVED (LEAN). The site labels the problem DISPROVED (LEAN) (page last edited 30 September 2025) and credits Jonas Barfield with the solution $10!=48^4-36^4$, recorded on its claim page; the Lean qualifier refers to the formal-conjectures file, which checks the witness by decide and which this corpus has not built. There is no refereed write-up. The standing in the frontmatter derives from the claim pages.

Source. erdosproblems.com/399, accessed 2026-09-04 and, with its empty discussion thread and proof-claims list, the community database and the formal-conjectures file, 2026-10-07. The site cites the problem from p. 77 of Erdős and Graham's 1980 problem book. Cite as: T. F. Bloom, Erdős Problem #399, https://www.erdosproblems.com/399.

References.

• [Br32] Breusch, Robert, Zur Verallgemeinerung des Bertrandschen Postulates, daß zwischen $x$ und 2 $x$ stets Primzahlen liegen. Math. Z. (1932), 505-526.
• [ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 77. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [ErOb37] Erdős, P. and Obláth, R., Über diophantische Gleichungen der Form $n!=x^p\pm y^p$ und $n!\pm m!=x^p$. Acta Litt. ac Sci. Reg. Univ. Hung. Fr.-Jos., Sect. Sci. Math. 8 (1937), 241-255. Library home: erdos_1937_uber_diophantische_gleichungen_der_form_und.
• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section D25 "Equations involving factorial $n$", printed p. 301: "Erdős & Obláth dealt with the equation $n!=x^p\pm y^p$ with $x\perp y$ and $p>2$". Library home: guy_2004_unsolved_problems_number_theory.
• [PoSh73] Pollack, Richard M. and Shapiro, Harold N., The next to last case of a factorial diophantine equation. Comm. Pure Appl. Math. 26 (1973), 313-325.

Formalization. The formal-conjectures file, linked at the commit of 18 September 2026, states the problem as erdos_399 : answer(False) ↔ … and proves it from the witness $(10,48,36,4)$ by decide. At that commit it also states, with sorry, the coprime theorem of Erdős and Obláth, the Pollack–Shapiro theorem on $n!=x^4-1$ and the two-squares classification, and proves Cambie's observation for $k=4$, a proof added on that date and linked from Cambie's claim page; at the commit of 12 January 2026 that the claim page pins for the main proof, that variant was still stated with sorry. The proof of the main statement is linked at both commits from the claim page. This corpus has not built it.

Current assessment

The question, as the site states it (page last edited 30 September 2025): is a factorial never a sum or difference of two $k$th powers with $k>2$, apart from the trivial cases with $xy=1$? The answer is no.

The resolution. $10!=48^4-36^4$, found by Jonas Barfield; the witness is checked by hand on the claim page, which also records the acceptance: the site's curator credits the solution, the community database records the problem as disproved with a Lean proof, and the formal-conjectures file checks the witness. There is no refereed write-up and no Lean file is built here. The bases $48$ and $36$ share the factor $12$; the solution lies exactly in the case the earlier results do not reach.

What was known for coprime bases. Erdős and Obláth [ErOb37] proved that $n!=x^p\pm y^p$ has no solution with $x,y$ coprime beyond the trivial one when $p\ge3$ is not a power of $2$, and handled differences with $p=8$ and so with every $p=2^\alpha$, $\alpha\ge3$; for $p=4$ they excluded coprime differences only for sufficiently large $n$ (Satz 3, p. 254), with the prime number theorem for the progressions modulo $4$, and sums of even powers fall to their two-squares argument. The site records their theorem as the coprime case with $k\ne4$; their claim page states what the paper proves. Pollack and Shapiro [PoSh73] are credited with the remaining case, in two accounts that differ. The site, followed by the formal-conjectures variant pollack_shapiro, says they showed that $n!=x^4-1$ has no solution. Erdős and Graham [ErGr80], the problem's source (p. 77), write that Erdős and Obláth settled $n!=x^k\pm y^k$ with $(x,y)=1$ and $k>2$ for $k\ne4$ and that Pollack and Shapiro showed that it also has no solutions for $k=4$, which would close the coprime case for every $k>2$; the paper's title, the next to last case of a factorial Diophantine equation, does not decide between the two readings. [PoSh73] is not held in the library, so which statement it proves is unconfirmed; the wider statement is recorded as the monograph's report and the narrower one as the site's, on their claim page. The site adds two observations. Erdős and Obláth noted that, by the theorem of Breusch [Br32] that consecutive primes $q_i<q_{i+1}$ congruent to $3$ modulo $4$ satisfy $q_{i+1}<2q_i$ beyond $q_1=3$, together with Fermat's two-squares theorem, $6!=12^2+24^2$ is the only solution of $n!=x^2+y^2$ with $xy>1$ (the condition excludes the trivial $2!=1^2+1^2$). Stijn Cambie observed that the sum $x^4+y^4$ of two coprime fourth powers, not both equal to $1$, is $1$ or $2$ modulo $8$ while $n!$ is divisible by $8$ for $n\ge4$, so $n!=x^4+y^4$ has no coprime solution with $xy>1$. Apart from sums of even powers, which the two-squares classification excludes coprime or not, none of these results constrains the non-coprime case with $k>2$, which is where the solution lives. The Erdős–Obláth and Pollack–Shapiro theorems are refereed partial results and Cambie's observation a pending one, each on its claim page; the two-squares classification concerns $k=2$, outside the problem, and has no page. Guy [Gu04] records the Erdős–Obláth theorem in section D25 and the two-squares remark under D2.

Search scope, 2026-10-07: the site's problem page, its discussion thread (no comments) and proof-claims list (none), the community database and the formal-conjectures file; the library's card for [ErOb37] for the coprime results, and the monograph [ErGr80] at p. 77 for the source's account of them. The exact date on which Barfield's solution was found or first posted is not recorded; the site's page carried it by 7 April 2025, the date of the earliest archived copy that does.

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.

• erdos_1937_uber_diophantische_gleichungen_der_form_und
• erdos_1937_uber_diophantische_gleichungen_der_form_und / equation_ia
• erdos_1937_uber_diophantische_gleichungen_der_form_und / satz_1
• erdos_1937_uber_diophantische_gleichungen_der_form_und / satz_2
• erdos_1937_uber_diophantische_gleichungen_der_form_und / satz_3
• guy_2004_unsolved_problems_number_theory