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

An application of the continued fractions for \FD... 11 Proof. If x = (1, k - l,2,fc - 1,1,4k) then x > 1, 1 x = 1 + 1 fc - 1 + — 1 2 + — 1 fc - 1 + — 1 1 + 1 4 k + x and so ij + - - 3) = 0 from which (using ~ > 0) we can see that 7(2

k + l)

2 - 4 = + x It is known that \/~D = yj(2k -f I)

2 — 4 = (a

0, ai, .. ., a

s ) where a

0 = A/D; = 2k, and yJ~D — clq -\ , where x = [a\

..... a

s). x Every irrational number can be expessed in exactly one way as an infinite simple continued fraction. Thus the first part of the lemma is proved. The proof of other three parts is carried out analogously. We can see these formulae in [6] p. 321., too. Proof of Theorem 1. We have only to apply the Lemma 4. and Lemma 7. By (22) y/D = s/{2k + l)

2 -4 = (2fc,MT- 1, 2, k - 1,1, 4k), k > 2 and so the representation of \f~D as a simple continued fraction has a period consisting of s ™ 6 terms. This terms are a

0 = 2k, ai = 1, a

2 = k - 1, a

3 = 2, a

n = k - 1, a

5 = 1, a

6 = 4k.