Erdős Problem 399
Statement.
Is $n!=x^k\pm y^k$ impossible for positive integers $x,y,n$ with $xy>1$ and $k>2$?
A counterexample to the stated question.
The answer is no, as the example attributed to Jonas Barfield on the problem page shows: \[
48^4-36^4
=12^4(4^4-3^4)
=20736\cdot175
=3628800
=10!.
\] Here $x=48$, $y=36$, and $k=4$, so every condition in the displayed question is satisfied. In particular, the literal universal assertion fails; it does not impose coprimality.
A proved restriction for a stronger variant.
For comparison, the fourth-power sum is impossible under the additional hypothesis $\gcd(x,y)=1$. This short obstruction is attributed to Stijn Cambie on the source page. If $n\geq4$, then $8\mid n!$. A fourth power is $0$ or $1\pmod8$. Coprimality excludes two even bases, so $x^4+y^4$ is $1$ or $2\pmod8$, a contradiction. If $n\leq3$, the condition $xy>1$ makes at least one base at least $2$, and $x^4+y^4\geq17>n!$. This proves exactly the claimed restricted case; it is not a classification of all coprime differences or all exponents.
Scope and source.
The single explicit difference settles the stated problem completely. The additional modular argument is included to distinguish the coprime variant. Source:
https://www.erdosproblems.com/399, checked 24 September 2026. Both arguments are reconstructions of known observations, with no novelty or formal-verification claim.
Completion estimate.
The original assertion is disproved by a fully checked integer identity.
COMPLETION: 100%