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

42 A. Grytczuk and T. Vorobév .(1) r 2 x Denoting by x^ = x

t - q\x^ and by A

2 = f j we obtain Mi V

A" Continuing this process we obtain in the last step the following matrices 0 \ / 0 x

{}

] o o

r U <i Consequently we obtain the following representation: <»> -;/)-Co !)•••(; f if rn is even, or if m is odd. Prom (26) we have det A — ciiXi + djXj = D — —dx (i) and we obtain d \ D. On the other hand putting x

k = v

k for k = 1, 2,.. ., n and k / i,j we have D = diXi + = b — ^T^ OfcVjfc. fc=i fci^t.j In similar way by (27) it follows that det A — D — dx^ and we obtain d I D. In both cases we have x^ = — ^ if 771 is even and x^ = ® if m is odd. Hence, from (27) and (26) we obtain (23)-(24) and the proof is complete. Consider the following equation: (28) 12x + 7y + 5z = 24.