Some conjectures on congruences

1. Notation. Let Z and N be the sets of integers and positive integers respectively. For b, c ∈ Z the Lucas sequences {U n (b, c)} and {V n (b, c)} are defined by (1.1) U 0 (b, c) = 0, U 1 (b, c) = 1, U n+1 (b, c) = bU n (b, c) − cU n−1 (b, c) (n ≥ 1) and (1.2) V 0 (b, c) = 2, V 1 (b, c) = b, V n+1 (b, c) = bV n (b, c) − cV n−1 (b, c) (n ≥ 1). Let d = b 2 − 4c. It is well known that for n ∈ N, (1.3) U n (b, c) = 1 √ d b+ √ d 2 n − b− √ d 2 n if d = 0, n(b 2) n−1 if d = 0 and (1.4) V n (b, c) = b + √ d 2 n + b − √ d 2 n. Let [x] be the greatest integer not exceeding x, and let (a m) be the Jacobi symbol. For m ∈ Z with m = 2 α m 0 (2 m 0) we say that 2 α m and m 0 is the odd part of m. For t ∈ Z let δ(t) = 1 if 8 | t, −1 if 8 t.