Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)

H.-MOLNÁR, S., Approximation by quotients of terms of second order linear recursive sequences of integers

Approximation by quotients of terms of second order linear recursive 25 (4) Therefore using the equality a = b, inequality (2) can be written in the form G

n-(-1 1 > C I H

r t. H

n = c\H

n\

k1\(bt-b

1ß)ß

r k-1 — c lacv I I 1 — cl Ka = c|a|

f

c1(|a|

f

c1|/?|)

n|6í-6i/?| \ß

n\ \bt - biß\ k-l Since L < 1 and a • ß — — B, this inequality holds for infinitely many n only if |/?|H

fc_

1 = \B\\a\

k~

2 < 1, that is if k < 2 - = k

0 and in the case k = k

0 we need 1 c < I a^o-^bt By (3) and by a = b it follows that \bt - b

xß\ = - b

xß\ = \a

xa - b

xß\ = |Gi| = \psB\. Therefore using the fact that a — ß = \/~D c < JD qsB ko-l \psB\ = Co Thus by (4) we obtain that (2) holds for infinitely many n if k < ko or k = ko and c < cq

. (If ^ > 0 then for any sufficiently large n, else for any sufficiently large even n.) References [1] Kiss, P., Zero terms in second order linear recurrences, Math. Sern. Notes (Kobe Univ.), 7 (1979), 145-152. [2] Kiss, P., A Diophantine approximative property of the second order linear recurrences, Period. Math. Hungar., 11 (1980), 281-287. [3] Kiss, P. AND SINK A, Zs., On the ratios of the terms of second order linear recurrences, Period. Math. Hungar., 23 (1991), 139-143. [4] Kiss, P. AND TICÍIY, R. F., A discrepancy problem with applications to linear recurrences I., Proc. Japan Acad. 65, No. 5 (1989) 135-138. [5] Kiss, P. AND TICHY, R. F., A discrepancy problem with applications to linear recurrences I., Proc. Japan Acad. 65, No. 6 (1989), 191-194.