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

8 Béla Zay Theorem 4. Let k (k > 1) be a natural number and D = k

2 + 1. Let a and ß denote the zeros of f^(x) = x

2 — 2kx — 1 with a > ß. Denote by M the set of positive solutions of x

2 - Dy

2 — N. (a) If N = I

2 and 1 < I < V~k then (b) If N = -I

2 and I < I < \fk then í I , / (a

2

m — ß

2

m) 1 M= Ux,

y):x = -(a

2

m +ß

2™), y= K ^ _ £ \ m> 1 . (c) If 1 < |A

r| < k and J TV | isn't a square of a natural number then M = 0. Proofs To prove Lemma 1. we need the following two lemmas. Lemma 5. Let f

nj

r

2 (^i ? x2i • • • •> xn) and g

n+

2 (^i

, x

2

l..., x

n) be the pohnomials which are defined by recurring relations fn+ 2(^1, • • - ,®n) = X

nf

n

+i(x

l, . . . ,X

n_i) + /

n(®l, • - - , a?n-2 ), TO > 1 and g

n+

2(x

u.

. .,x

n) = xig

n+i(x

2

i...,x

n) + g

n(x

3,..., x

n), n> 1 respectively, where /1 = 0i =0 and f

2 = g

2 = 1. Then fn+ 2(^1, = 071+2

(si, • TO > -1 aiso holds. Proof. We can easily verify that /l =01, /2 = 02, /3(^1 ) = «1 = 03^1 ) and U{xi,X

2) = x

2f

3(xi) + /2 - 03(^2)^1 +02 = 54(^1,^2)Assume that n > 3 and /n+2-i(^l 5 • • • ) ^n-l ) — 071+2-^(^17 • • • 5 ^n-l ) holds for i = 1,2,3,4.