The equation x_n−1/x−1= y_q with x square

Abstract

It has been conjectured that equation formula here has only finitely many solutions. We observe that (x, y, n, q)=(3, 11, 5, 2), (7, 20, 4, 2) and (18, 7, 3, 3) are solutions of (1). These are the only known solutions and perhaps (1) has no other solution. Ljunggren [8] proved in 1943 that (1) with q=2 has no solution other than x=3, y=11, n=5 and x=7, y=20, n=4. Shorey and Tijdeman [16] confirmed the conjecture if x is fixed. Let z>1 be an integer. The main purpose of this paper is to show that (1) has no solution if x is restricted to the infinite set of squares z² with z[gt-or-equal, slanted]32 and z[set membership]{2, 3, 4, 8, 9, 16, 27}.