Effective bounds show q ≤ log₂(b+1) for (2^k-1)(b^k-1)=y^q except for a finite list of exceptions when 3≤b<10^6, and no solutions exist for the equation with exponent n when b is in {5,7,11,13,21,23,27,29}.
Szalay,On the Diophantine equation(2 n −1) (3 n −1) =x 2, Publicationes Mathematicae Debrecen,57(1-2)(2000), 1–9
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The diophantine equation $(2^{k}-1)(b^{k}-1)=y^{q}$
Effective bounds show q ≤ log₂(b+1) for (2^k-1)(b^k-1)=y^q except for a finite list of exceptions when 3≤b<10^6, and no solutions exist for the equation with exponent n when b is in {5,7,11,13,21,23,27,29}.