Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)
ZAY , B., An application of the continued fractions for ... in solving some types of Pell's equations
4 Béla Zay Auxiliary results The purpose of this paper is to give such second order linear recurrences in case of |A r| < y/D and in some special cases. We shall use a lemma of P. KlSS [3] and some theorems from [5] and [6]. Let 7 be a real quadratic irrational number and let (3) 7 = (ao, öi, Ű2,. ..) = (ao,cii,.. . . . ,a i+ s_i) be the representation of 7 as a simple periodic continued fraction, where s is the minimal period length of (3). P. Kiss proved: If the 72-th convergent to 7 is jf- = (ao ? > • • • ? an) and the n-th convergent to 70 = (a f,... , a t+ s1) is = (a t,a t +i,. . ., a i+ n), then (as it was proved by P. Kiss [3]) (4) //( n + 2) s+ r = (/l s-1 + k s-2)H(n+l)s+r + (~1) S + 1 H-ns+r, and (5) Ä( n + 2)s+r = (hs-l + ^s-2)Ä(n+l)s+r + 1) S+ 1 A ns+ r, where 72 > 0, r = 0,1, . . . , s — 1 and we assume, that k-1 = 0. In the special case of 7 = y/D we prove the following lemma. Lernma 1. Let D be a positive integer which is not a square of a natural number and let (6) VD = {a 0 , ai,...,a s) be the representation of y/D as a simple continued fraction expansion, where s is the period length of (6). If the n-th convergent to y/D is —— = (a 0, ai,..., a n) •t*- n then (7) #(n + 2)s+r = 2// s_i /7( n + 1) s + r + (-1 Y + 1 H n s+ r and (8) /i( n + 2) s+r = 2i/ s_i K {n + l)s+ r + (-1 Y + lHns +r for every integer n > 0 and r (0 < r < s - 1).
