Near-squares in binary recurrence sequences.

We call an integer a near-square if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers a ≥ 3 by u 0 (a) = 0 , u 1 (a) = 1 and u n + 2 (a) = a u n + 1 (a) − u n (a) for n ≥ 0. We show that for a given a ≥ 3 , there is at most one n ≥ 5 such that u n (a) is a near-square. With the exceptions of u 6 (3) = 1 2 2 and u 7 (6) = 2 3 9 ⋅ 1 3 2 , any such u n (a) can be a near-square only if a ≡ 2 (mod 4) , n ≡ 3 (mod 4) is prime and n ≥ 1 9. This is a part of a more general phenomenon regarding near-squares in nondegenerate recurrence sequences defined for the integers a and b = − b 1 2 by u 0 (a , b) = 0 , u 1 (a , b) = 1 and u n + 2 (a , b) = a u n + 1 (a , b) + b u n (a , b) for n ≥ 0. This arises from a novel Aurifeuillean-type factorization of u 2 n + 1 (a , b) we have found. [ABSTRACT FROM AUTHOR]