ON THE DIOPHANTINE EQUATION x2 dy4 = 1 WITH PRIME DISCRIMINANT II BY

Abstract. Let p denote a prime number. P. Samuel recently solved the problem of determining all squares in the linear recurrence sequence fTng, where Tn and Un sat-isfy T 2n pU2n = 1. Samuel left open the problem of determining all squares in the se-quence fUng. This problem was recently solved by the authors. In the present paper, we extend our previous joint work by completely solving the equation Un = bx 2, where b is a xed positive squarefree integer. This result also extends previous work of the second author. 1. Introduction. Ljunggren [5] proved that the Diophantine equation (1) X2 dY 4 = 1 has at most two solutions in positive integers, and gave precise information on the location of the solutions when two solutions exist. This general the-orem has recently been improved substantially in [11]. Specically, in tha