The Rocky Mountain Journal of Mathematics 33(2003), no.3, 1123-1145. VALUES OF LUCAS SEQUENCES MODULO PRIMES

Let p be an odd prime, and a, b be two integers. It is the purpose of the paper to determine the values of u(p±1)/2(a, b) (mod p), where {un(a, b)} is the Lucas sequence given by u0(a, b) = 0, u1(a, b) = 1 and un+1(a, b) = bun(a, b) − aun−1(a, b) (n > 1). In the case a = −c2, a reciprocity law is established. As applications we obtain the criteria for p | u p−1 4 (a, b) (if p ≡ 1 (mod 4)) and for k ∈ Q0(p) and k ∈ Q1(p), where Q0(p) and Q1(p) are defined as in [10]. 1.Introduction. Let a and b be two real numbers. The Lucas sequences {un(a, b)} and {vn(a, b)} are defined as follows: (1.1) u0(a, b) = 0, u1(a, b) = 1, un+1(a, b) = bun(a, b)− aun−1(a, b) (n > 1); (1.2) v0(a, b) = 2, v1(a, b) = b, vn+1(a, b) = bvn(a, b)− avn−1(a, b) (n > 1). It is well known that (1.3) un(a, b) = 1 √ b2 − 4a (b +√b2 − 4a 2 )n − (b−√b2 − 4a 2 )n)