EASY CRITERIA TO DETERMINE IF A PRIME DIVIDES CERTAIN SECOND-ORDER RECURRENCES

Let F(a, b) denote the set of all second-order recurrences w (a, b) satisfying the recursion relation Wn+2 = aW(n+1) + bw(n), where the discriminant D = a(2) + 4b and a, b, w(0), and w(1) are all integers. Let u(a, b) denote the recurrence with initial terms u(0) = 0 and u(1) = 1. We say that the prime p is a divisor of w(a, b) if p vertical bar w(n) for some integer n >= 0. Let z(p) denote the least positive integer n such that u(n) equivalent to 0 (mod p). Then z(p) vertical bar p - (D/p), where (D/p) denotes the Legendre symbol. Define the index i(p) as i(p) = p - (D/p)/z(p). When i(p) = 1 or 2, we will find easy criteria to determine exactly when p is a divisor of w(a, b) based on the residue class or quadratic character of w(1)(2) - aw(1)w(0) - bw(0)(2) modulo p. This generalizes results of Vandervelde when a = b = 1.