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Claim. M. A. Bennett and S. Siksek, A conjecture of Erdős, supersingular primes and short character sums, Ann. of Math. (2) 191 (2020), no. 2, 355--392 (preprint arXiv:1709.01022, 4 September 2017). Theorem 2 (printed p. 357) reads: "There is an effectively computable absolute constant $k_0$ such that if $k\geq k_0$ is a positive integer, then any solution in integers to equation (2) with prime exponent $\ell$ satisfies either $y=0$ or $d=0$ or $\ell\leq\exp(10^k)$." Equation (2) is $n(n+d)\cdots(n+(k-1)d)=y^\ell$ with $\gcd(n,d)=1$. A positive progression has $d\ne0$ and $y\ne0$, so for every $k\ge k_0$ the product is never an $\ell$th power with $\ell$ prime and $\ell>\exp(10^k)$, nor any power whose exponent has such a prime factor. The constant $k_0$ is effective but not computed. With Faltings's theorem the paper also gets finitely many solutions for each such $k$, which settles no further instance. Library home: bennett_2020_conjecture_erdos_supersingular_primes_short.

Covers. Every length $k\ge k_0$, every $d$, and every exponent with a prime factor $\ell>\exp(10^k)$, for Problem 672. Not covered: smaller exponents, and lengths below $k_0$.

Depends on. Nothing in this wiki; the result rests on the cited paper.

Acceptance. Refereed: Annals of Mathematics (2) 191 (2020), no. 2; the Crossref record dates the issue to March 2020. The page is named by the arXiv posting of 4 September 2017. The site's commentary credits the theorem, but the site labels the problem VERIFIABLE, an open label, so the commentary is not reviewed evidence.