Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

GRYTCZUK, A. and VOROBEV, N. T., On some applications of 2 X 2 integral matrices

38 A. Grytczuk and T. Vorobév On the other hand by an easy calculation, from (3) we obtain <*> (n ::)=(! i;)(? SMS *)• By (5) it follows that xq = 8, yo = 14. In 1986 A. J. van der Poorten [3] observed that if (? J)(i ä)-(i o) = (L":::;)' ^

0'

1then — = [co;ci,...,c

n]. <7n Based on this observation he gave many interesting applications to the theory of continued fraction and also to the description of the solutions of the well-known Pell's equation x

2 — dy

2 = 1. In [2] we gaves some connections between fundamental solution (xo,yo) of the Pell's equation and representation of 2 X 2 integral matrix as a product of powers of the prime elements in the unimodular group. In the present paper we give such connections between the fundamental solution (xo,yo) of the non-Pellian equation x

2 — dy

2 = —1 and the corresponding matrix representation. We prove the following: Theorem 1. Let \fd. — [go

; <7i

, • • •, Qs] 5 d > 0 and s > 1 is odd is odd, be the representation of y/d as a simple periodic continued fraction. Then the fundamental solution (zo, yo) of the non-Pellian equation (6) x

2 — dy

2 = — 1 in contained in the second column of the following matrix: 0

:)"(: :)"• •(: ;)" Proof. First we prove that if k = 2n, n = 1,2,..., then ^ f

1 »V

1 f

1 ^\ = {p

k1 Pk\ [ ) V° WW lJ 1J \Qk-i Qk)