A diophantine equation concerning finite groups (Q1901074)

The author proves that all solutions \((y,m,n)\) of the equation \[ 3^m - 2y^n = \pm 1,\;y,m,n \in \mathbb{N},\;y > 1,\;m > 2,\;n > 1, \tag{*} \] satisfy \(y < 10^{6 \cdot 10^3}\), \(m < 1.4 \cdot 10^{15}\) and \(n < 1.2 \cdot 10^5\). The proof uses a lower bound for a linear form in two logarithms by \textit{M. Laurent} [Appendix of ``Linear independence of logarithms of algebraic numbers (M. Waldschmidt), IMS Report No. 116 (Madras, 1992; Zbl 0809.11038)]. In a forthcoming paper, \textit{L. Yu} and the author show that \((*)\) has only the solution \((y,m,n) = (11,5,2)\). A corrected statement of the result of \textit{P. Crescenzo} [Adv. Math. 17, 25-29 (1975; Zbl 0305.10016)] is also given.