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

#4n + 2 — ^4n + 2 = An application of the continued fractions for \FD... 13 (a - 2)g

2n+

1 (ß — 2)ß

2n+

l 2 (g - 2)g

2n+

1 - (/3 - 2)/j

2n+1 2(a-ß) a2n+2 ß2n + 2 tt

2

n+

2 - /3

2

n+

2 //

4n + 3 = - , K\

n

+3 = — —— 2 2(q - ß) for n > 0 and (2a — l)g

2

n+

2 + (2/3 — l)ß

2n+

2 n 4 — 4n + 4 — ^4n-f4 = 2 (2a - l)a

2

n+

2 - (2ß - l)/?

2

n+

2 2(a-/5) for n > — 1 and f 1 for r = 3 \ Hi - DK

2

n = j 4, for r = 1 I = (-1 y1 c

r+

l ,n> 0. I 5 — for r — 2 or 4 j Proof of Theorem 3. VD = \/A:

2 - 1 = (k - 1,1.2* - 2), for fc > 2 _ g

n+

1 + /5

n+

1 , g

n+

1 - ß

n+

l #2n+l — Ö j ^2n+l — #271+2 — &2n+2 = 2 7 <*-/? (a - l)g

n+

2 + (ß - l)ß

n+2 2 (g - l)g

n+

2 - (ß - i)ß

n+

2 a- ß for n > — 1 and /w-= {

2fc

) £;:£}= (-ir'c

r+

1,

»

>o. Proof of Theorem 4. VD = y/k

2 + 1 = (Jfc, 2/c), for k > 1 g n+1 + ßn+1 _ _ ßn+1 H

n — _ 1 An'— a — ß