Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

JONES, J. P. and Kiss, P., Some congruences concerning second order linear recurrences

Some congruences concerning second order linear recurrences JAMES P. JONES and PÉTER KISS* Abstract. Let U

n and V

n (n=0,l,2,...) be sequences of integers satisfying a second order linear recurrence relation with initial terms U

0= 0, Ui=l, V

0=2, V

x =A. In this paper we investigate the congruence properties of the terms U

nk and V^*, where the moduli are powers of U

n and V„. Let U

n and V

n (n — 0,1, 2,...) be second order linear recursive sequences of integers defined by Un = AU

n-i - BU

n-2 {n > 1) and Vn = AV

n-i - BV

n2 [n > 1), where A and B are nonzero rational integers and the initial terms are Uq = 0, Ui = 1, V

0 = 2, V\ — A. Denote by a, ß the roots of the characteristic equation x

2 - Ax B = 0 and suppose D = A

2 - 4i? / 0 and hence that a ^ ß. In this case, as it is well known, the terms of the sequences can be expressed as a

n - 3

n (1) U

n = — and V

n = a

n + /T a - ß for any n > 0. Many identities and congruence properties are known for the sequences U

n and V

n (see, e.g. [1], [4], [5] and [6]). Some congruence properties are also known when the modulus is a power of a term of the sequences (see [2], [3], [7] and [8]). In [3] we derived some congruences where the moduli was I7

3, V

2 or V

3. Among other congruences we proved that U

n

k = kB

n^U

n (mod/7

3) * Research supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA T 16975 and 020295.