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In Problem 399, the equation $n!=x^4+y^4$ has no solution with $\gcd(x,y)=1$ and $xy>1$. A fourth power is $0$ or $1$ modulo $8$ as its base is even or odd, and coprime $x,y$ are not both even, so $x^4+y^4$ is $1$ or $2$ modulo $8$, while $8\mid n!$ for $n\ge4$; for $n\le3$, $n!\le6<16\le x^4+y^4$.

Covers. The sum $n!=x^4+y^4$ with $\gcd(x,y)=1$ and $xy>1$. The same case, without the coprimality condition, also follows from the two-squares argument in the introduction of Erdős and Obláth (see their claim page): $n!$ is a sum of two squares for no $n\ge7$, and $6!=12^2+24^2$ has no representation as a sum of two fourth powers.

Depends on. No page of this wiki.

Standing. Claimed. The remark is credited to Stijn Cambie in the site's commentary; it is absent from the archived copy of the site's page of 7 April 2025 and present in the copy of 9 September 2025, the date this page carries. The site's label settles the problem through Barfield's counterexample and not through this remark, so reviewed is not listed, and the remark has no written source of its own. The formal-conjectures file, at the commit of 18 September 2026 linked above, proves erdos_399.variants.cambie from this argument; the contribution says it formalizes Cambie's argument as the docstring states it and was developed with Claude. It is third-party Lean that this corpus has not built or audited, so formalized is not listed.