Some results concerning the exponential Diophantine equation (an-1)(bm-1) = x2

Let a, b > 1 be two given unequal integers. In this paper, we examine, under some assumptions on the residue of a modulo an odd prime factor p of b, the Diophantine equation of the title with positive integer unknowns n, m and x. For example, we show that it has no solution when p = +/- 3 (mod 8), a = p-1 (mod 2 p) and b = 3 (mod 4). We also give a necessary and sufficient condition for the existence of solutions with n = m, when p = +/- 3 (mod 8), a = -1 (mod p) and when p = 17, a(2) = -1 (mod 17).