Difference between powers and squares of integers
Inspired by \textit{J. H. E. Cohn}'s conjecture [Acta Arith. 65, 367-381 (1993; Zbl 0795.11016)] the author establishes that for any solution in positive integers \(x,y,n\) of the diophantine equation \[ x^2+7=y^n, \] either \(x\in\{1, 3, 5, 11, 181\}\), or \(y>10^9\) and all prime factors of \(n\) exceed 5000. For fixed values of \(n\leq 5000\), the author uses the classical Thue equation approach (factorization and congruence considerations), and for fixed value of \(y\leq 10^9\) results from the theory of linear forms in logarithms are applied.
scientific article; zbMATH DE number 1247928