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Claim. Let $S$ be the set of primes $p\equiv5\pmod 8$. Theorem 1.2 of W. Ma, An elementary note on three consecutive powerful numbers, arXiv:2608.23418 (2026-08-24): there are no three consecutive powerful numbers $x^n-1=q_1^3y^2$, $x^n$, $x^n+1=q_2^3z^2$ with $q_1,q_2$ prime, $x,y,z$ integers and $n\geq5$ an integer all of whose prime factors lie in $S$. Corollary 1.3 deduces that, for such $n$, the equation $(2ax)^{2n}-1=q_1^3q_2^3y^2$ has no solution in integers $a,x,y$ and primes $q_1,q_2$. The paper extends Chan's theorem, whose middle member is a cube, to middle members that are such $n$th powers, using results of Nagell and Ljunggren, Lebesgue and Ko.

Covers. Triples whose middle member is an $n$th power for an exponent $n$ as above and whose outer members are both a prime cubed times a square, a partial no to the question of Problem 364.

Standing. The claim is pending: an arXiv preprint, not refereed, and the site neither lists it on its proof-claims tab nor credits it, so there is no acceptance evidence.