Glasnik Matematicki, Vol. 38, No.1 (2003), 11-18.

NEAR SQUARES IN LINEAR RECURRENCE SEQUENCES

P. G. Walsh

Abstract.   Let T > 1 denote a positive integer. Let U_n denote the linear recurrence sequence defined by U₀ = 0, U₁ = 1, and U_k+1 = 2TU_k - U_k-1 for k ≥ 1. In recent years there have been some improvements on the determination of solutions to the Diophantine equation U_n = cx², where c is a given positive integer. In this paper we use a result of Bennett and the author to determine precisely the integer solutions to the related equation U_n = cx² ± 1, where c is a given even positive integer.

2000 Mathematics Subject Classification.   11D25, 11B39.

Key words and phrases.   Linear recurrence sequence, diophantine equation, Pell equation.