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

32 James P. Jones and Péter Kiss After some calculation (3), (4) and (5) imply (6) U

n{k

+

4) EE U

nT + U*s (mod D

2Ul), where Jfe+3 (k-M)-l T = (2

(k + 2) - k)B~

n = (k + 4)B —

2

— and fc±i (A; + 2) ((A; + 2)

2 - 1) fc

+i 5 = (k + 2)DB~

n + 2 MV 1 -DB* _ k(k

2 - 1

) *±i

n = (A

+ 4) ((fc + 4)

2 - 1

) (Li^n 24 24 and so by (6), ^n(Jt+4) = (* + 4)B

{±

±^U

n (k + 4) ((/c + 4)

2 - l) (fc

+4)3 , , , . + V 2

4 DB-^~

nUl (mod D

2Ul). Hence the congruence holds also for k + 4 and for any odd positive integer k. The other congruences in the Theorem can be proved similarly using Lemma 1, 2, 3 and the identities U2n — V

nU

n

, V

2

n = V,I — 2B

n = 2B

n + DU

2, Uzn = U

nV

2 - B

nU

n

, F

3

n - V

n

3 - 3B

nV

n = B

nV

n + DV

nU

2, Utn = U

nV* -2B

nU

nV

n, F

4

n = V* - AB

nV

2 + 2B

2

n

. References [1] D. JARDEN, Recurring sequences, Riveon Lematematika

, Jerusalem (Israel), 1973. [2] J. P. JONES AND P. KLSS, Some identities and congruences for a special family of second order recurrences, Acta Acad. Paed. Agriensis, Sect. Math. 23 (1995-96), 3-9.