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

On some applications of 2x2 integral matrices 43 We have (12,7,5) = 1. Equation (28) can be represented in the form 7y + 5z = 24 - 12x = 12(2 - a); x = a. On the other hand, we have: \ = [l; 2,2]. By the Theorem 2, we have: A = 7 ~z\ _ (

1 lWl oVfl 1\V0 —(24 — 12a) \ 5 y ) V 0 V V

1 1/ LJ U 0 J where Z) = det A = 24 — 12a, d = (7, 5) = 1, thus d j D. So we obtain 4 _ f 7 _ ( 3 l\ (2 -(24-12a) ^ _(l -3(24-12a) ^ by) V2 1 / V

1 0 7 V

5 —2(24 — 12a) J and we have x = a, y = -2(24 - 12a), 2 = 3(24 - 12a), where a is an arbitrary integer. References [1] A. GRYTCZUK, Application of integral matrix ^ ^ J to the determination of integer solutions of the equation ax — = ±1, Biul. WSInz. Mat.-Fiz. N^ 4., (1970), Zielona Góra, 149-153, (in Polish). [2] A. GRYTCZUK and N. T. VOROB'EV, Apphcaiton of matrices to the solutions of Diophantine equations, Vitebsk, Bielyorussia, (1990), (pp. 44), (in Russian). [3] A. J. VAN DER POORTEN, An introduction to continued fractions, London Math. Soc. Led. Note Ser. N^ 109., (1986), 99-138.