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
40 A. Grytczuk and T. Vorobév On the other hand by (13) and (15) we get (16) detFo = 1 = P S2QS-i - PS-IQS-2Since (17) P s_ 1 = qoQs-I + QS-2 and DQ S_ 2 = qoP S-I + P 82, by (17) we have (18) P s 2_! - i/g 2 s_l = Ps-LQS-2 - PS-2QS- 1. On the other hand it is well-known that (19) P s-iQ s-2 = (-l) s. Since s > 1 and 5 is odd then by (18), (19) and (16) we obtain (20) PU - dQl_ x = -1, so (xo,yo) = (P s_i,Q s_i) and the proof is complete. For example consider the following non-Pellian equation: x 2 - 13 y 2 = -1. We have — [3; 1,1,1,1, 6] and q 0 = 3, q X = q 2 = <?3 = g 4 = 1, 95 - 6; 5 = 5 is odd. Then by the Theorem 1 we have 1 1\ 3 (l 0 Wl l\ {I 0\ { I 1 F[ ) 'o 1 j \i 1 j vo íyv 1 17 V 0 1 1 3\/l l\ (l 1 \ / 4 7 \ (1 l\ _ ill 18 o i)\i 27 V 1 2/~ V 1 2;vi 2 / ~ v 3 5 and consequently xq = 18, yo = 5. Now, we gave a possibility for an application of 2 x 2 integral matrices to the examination of the equation: (22) a\X\ +02^2 -f h a nx n - b. Namely, we prove the following:
