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Lehmer sequences have many interesting properties and often arise in the study of Diophantine equations. The arithmetic properties of the numbers Pn can be found in [5, 15]. The main purpose of the present paper is to investigate the occurrence of squares in Lehmer sequences and their applications to Diophantine equations of the form (1.2) aX − bY 2 = 2, where a and b are given positive odd integers. This type of problem has received considerable interest (see [3, 4, 11, 10, 14]). In certain ways it actually goes back to the classical work of Ljunggren [6–9], who was able to prove many theorems on equations of the form aX4 − bY 2 = c with c ∈ {±1,−2,±4}, but he did not prove any result on the case c = 2 (curiously). Therefore, the result of this paper can be viewed as a case that Ljunggren missed, for reasons that will never be known. Here as well as throughout the paper, we use ( A B ) to denote the Jacobi symbol of A with respect to B, where A and B are coprime integers. Rotkiewicz proved the following two results concerning the equations Pp = px2, Pp = x2, where p is an odd prime.

Squares in Lehmer sequences and the Diophantine equation Ax4-By2=2

Squares in Lehmer sequences and the Diophantine equation Ax⁴-By²=2