On Cohn's conjecture concerning the diophantine equation¶x2+ 2m = yn

Abstract.

In the past fifty years and more, there are many papers concerned with the solutions (x,y,m,n) of the exponential diophantine equation $x^2 + 2^m = y^n, x, y, m, n \in \mathbb{N}, 2 \not|\, y, n > 2$, written by Ljunggren, Nagell, Brown, Toyoizumi, Cohn and the others. In 1992, Cohn conjectured that the equation has no solutions (x, y, m, n) with m > 2 and $2 \mid m$. In this paper, using a quantitative result of Laurent, Mignotte and Nesterenko on linear forms in the logarithms of two algebraic numbers, we verify Cohn's conjecture. Thus, according to known results, we prove that the equation has only three solutions (x, y, m, n) = (5, 3, 1, 3), (7, 3, 5, 4) and (11, 5, 2, 3).