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Erdős and Obláth restrict to coprime $x,y$ and prove three theorems. Their Satz 1: apart from $2!=2$, no factorial is a sum or a difference of the $p$th powers of two coprime numbers when $p\ge3$ is not a power of $2$. Their Satz 2: a difference of the eighth powers of two coprime integers is never a factorial, which excludes differences for every $p=2^\alpha$ with $\alpha\ge3$. Their Satz 3, proved with the prime number theorem for the progressions $4k+1$ and $4k+3$: for sufficiently large $n$, $n!$ is not a difference of the fourth powers of two coprime integers; no threshold is given. For sums, the introduction reduces the equation to prime exponents and shows that $n!$ is a sum of two squares for no $n\ge7$, since some prime $q\equiv3\pmod4$ with $n/2<q\le n$ divides $n!$ exactly once; $6!=12^2+24^2$. The exponents not covered by Satz 1 are the powers of $2$, so for sums with $k>2$ even this reduction applies, coprime or not, once the small cases are checked: $720$ is not a sum of two fourth powers.

Covers. No solution of $n!=x^k\pm y^k$ in Problem 399 with $\gcd(x,y)=1$, $xy>1$ and $k>2$, except possibly a difference $n!=x^4-y^4$ with $n$ below the unspecified threshold of Satz 3. The site records the result as the coprime case with $k\ne4$. Nothing here constrains the case $\gcd(x,y)>1$ with an odd exponent or a difference, where the solution $10!=48^4-36^4$ lies.

Depends on. No page of this wiki.

Acceptance. Refereed: P. Erdős and R. Obláth, Über diophantische Gleichungen der Form $n!=x^p\pm y^p$ und $n!\pm m!=x^p$, Acta Litt. Sci. Szeged 8 (1937), 241–255; library card erdos_1937_uber_diophantische_gleichungen_der_form_und. The site's curator credits the coprime theorem with $k\ne4$ to this paper in the commentary, but the site's label settles the problem through Barfield's counterexample and not through this result, so reviewed is not listed. The formal-conjectures file states the site's version of the theorem as erdos_399.variants.erdos_oblath with sorry, which is not a formalization.