Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

JONES, J. P. and Kiss, P., Some identities and congruences for a special family of second order recurrences

4 lames P. Jones and Péter Kiss The sequences X{n

) and Y(n) have important applications to Diophantine equations and Hilbert's tenth problem since they give all solutions to the polynomial identity - (a

2 - 4) y' = 4 (see [4]). These sequences are special cases P = a and Q = 1 of more general linear recurrent sequences V

n and U

n of Lucas which was defined by the recursion U

n = PU

n-\ — QU

n2- Consequently many identities and congruence properties are know for our sequences X, Y and also for more general sequences (see, e.g. [1], [2], [3] and [5]). For example it is well known that Y(kn

) = 0 (mod F(n)) for any natural numbers k and n. Lucas [3] also showed many properties of these sequences, e.g. he showed that Y(2n) = X(n)Y(n) and X{2n) — X{n)

2 — 2 and so X(2n) = -2 (mod X(n)

2). The purpose of this paper is to prove some congruences involving Y(kn

) and X{kn

), where the modulus is a power of the n

t

h term. In the proofs we use formulas of (1) but sometimes we give other methods not using the Binet formula. Specifically we prove the following congruences: Theorem 1. Let A; be an even positive integer. Then Y(kn) = ^y(2n) (mod F(n)

3) for any integer n > 0. Theorem 2. Let A; be an odd positive integer. Then Y(kn) = kY(n) (mod Y(nf) for any integer n > 0. Theorem 3. Let A; be an odd positive integer. Then X(kn) EE (mod X(n)

2) for any integer n > 0. Theorem 4. Let A; be an even positive integer. Then X(kn) = 2{-l)

k/

2 (mod X(n)

2)