On the Diophantine equation $x^2 + 7^\alpha \cdot 11^\beta = y^n$

Abstract

In this paper, we give all the solutions of the Diophantine equation x 2 C 7˛ 11ˇD y n ; for the nonnegative integers˛;ˇ; x; y; n 3, where x and y coprime, except when˛:x is odd andˇis even.

Complete solution of the Diophantine Equation $x^{2}+5^{a}\cdot 11^{b}=y^{n}$

“…Several papers deal with the Diophantine equation (1.1) when S contains at least two distinct odd primes. Thus, all solutions of the Diophantine equation (1.1) were given in [2] for S = {5, 13}, in [28] for S = {5, 17}, in [30], [31] for S = {7, 11} -except for the case when ax is odd and b is even-, in [6] for S = {11, 17}, in [17] for S = {2, 5, 13}, in [9] for S = {2, 3, 11}, in [15] for S = {2, 5, 17}. In [27], Pink gave all the non-exceptional solutions of the equation (1.1) (according to the terminology of that paper) for S = {2, 3, 5, 7}.…”