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

30 James P. Jones and Péter Kiss when k is odd and a similar congruence for even k. In this paper we extend the results of [3]. We derive congruences in which the moduli are product of higher powers of U

n and V

n. Theorem. Let U

n and V

n be second order linear recurrences defined above and let D = A

2 - AB be the discriminant of the characteristic equation. Then for positive integers n and k we have 1. U

n

k=kB^T

L

nU

J

l + k{

kl~

l

) DB

h^

l

nUl (mod D

2U*)

} k odd, 2. U

n

k = $B

äT

l

nV

nU

n + ki

kl-*

) DB

L*

±

nV

nuZ (mod D

2V

nU

5

n), k even, 3. V

n

k =k(-l) B V

n + kl

kl~

l

> (-1 (mod V„

5), k odd, 4. V

n

k=2(-l)^ + B

Li

l

nV

2 (mod V*), k even, 5. (mod U

nV

*), k odd, 6. U

n

k = t(-l)

ä^B

ä^

l

nU

nV

n + k

<'

klf*

){-l)^B

!^

±

nU

nV* (mod U

nV£)

f k even, 7. V

n

k=B

k^

nV

n + *^±DB

kT

l

nV

nUl (mod D

2V

nU*), k odd, 8. V

n

k=2B^

n + ^-B

ä^

l

n DU

2

n (mod D

2U*), k even. We note that the congruences of [3] follow as consequences of this theorem. For the proof of the Theorem we need some auxiliary results which are known (see e.g. [6]) but we show short proofs for them. In the followings we suppose that A > 0 and hence that A + yJ~D , _ A-VD « = 2 ß

= 2 ' so that a - ß = y/~D, a + ß - A, aß = B and hence by (1) a

n - Q

n (2) u

n = -JLemma 1. For anj integer n > 0 we have U

3

n = 3U

nB

n + DUl Proof. By (2), using that a/3 — we have to prove that a

3

n-ß

3

n a

n - ß

n . mn n/a"-/?

nN

3 a

n - 3

n fa

n - Q

n\