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

6 Béla Zay be the representation of \fD as a simple continued fraction. Suppose that N is a non-zero integer with | TV j < \fD, and let (11) H-1 = 1, H

0 = a

0, H

m ~ a

m/r

m_i + #

m_

2, 1 < m < 2s

, (12) A'-i = 0, A'o = 1, K

m = ö

mÄ'

m_i + Jf

m_

2, 1 < m < 2s, (13) //(n + 2)H-r = 2i/

s_

1 tf(

n + 1)s+

r + (-l)

S+

1 #

ns+

r, 1 < r < 5, 71 > 0, (14) K

{

n+

2)s+

r = 2tf

s_

1A'

(n+1)s+

T. + (-l)

s+

1 K

ns+

r, 1 < r < 5, n > 0, and (15) c

ns+r+

1 = (-l)

n'

+

rl(fT

r

2 - DK

2

r), l<r<s. If 1 < r < 5, c

r

+i ^ 0 and \J

^ —— is a naturai number then let d

r = V "V*^

1 Denote by M the set of positive solutions (x, y

) of a;

2 - Dy

2 — N. Then (16) M = a: = d

rH

ns+

r, y = d

rK

ns+

r, n > 0, 1 < r < s}. This also means that: If there exists no natural numbers d

r (1 < r < s) which satisfy the above conditions then there isn't integer solution x = u, y - t of x

2 - Dy

2 = N (|JV| < D), that is M is the empty set. Theorems Applying Lemma 4. for some special equations we obtain the following results. Theorem 1. Let k (k > 2) be a natural number with D = (2k +1)

2 -4. Let a and ß denote the zeros of /i(x) = x

2 - (2k + l)x + 1 and let a > ß. Denote by M the set of positive (x, y) solutions of x

2 — Dy

2 = A

7. (a) If N — 41

2 and l(l<l< J\] is a naturai number

, then f am _Qm } M= |(x,y): x = l(a

m + ß

m), y = I —

—, m > 1 j (b) IfN = (21 - l)

2 and 1 < / < \ + ^ V 1 ) (a

3m+

3 +ß

3m+

3) \ a

3

m+

3 _ ß3m + 3 ~ß y = (

l - - ) :—

^ > m ^ 1