On Cohn's conjecture concerning the Diophantine equation \(x^2+2^m=y^n\) (Q1347771)

In 1992, J.H.E. Cohn conjectured that the Diophantine equation \(x^2+2^m=y^n\) has no solution \((x,y,m,n)\) with \(m\) even, \(m\), \(n>2\) and \(y\) odd. The author proves this conjecture. This implies that this equation has only the solutions \((5,3,1,3)\), \((7,3,5,4)\) and \((11,5,2,3)\). The main tool is an estimate on linear forms in two logarithms of algebraic numbers, but the proof combines also results on several classical Diophantine equations.