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 41 Theorem 2. Let (ai

, a.2,

..., a

n) = 1 and d = (a^aj) for some i,j E {1, 2,. .., n}, where (ai

, <22 5 • • •, an) denote the greatest common divisor of d\

, 02 i • • • j an £ Z

. Then the integer solutions of (22) are of the form: (Vl, t>2, • • • , • • • , ..., where CC 2 ^ X j 3X6 detemrined by the following matrix equalities: *> C; ?)•(!;)"(; :)"•••(; ;rc -.*)• if m is even and m if rn is odd, where ^ = [qo; q\

,. .., q

m], d | D and D = b — ^ a^v^

c. J fc=i Proof. Let (a;, aj) = d. We can assume without loss of generality that a; > aj > 0. Applying to a

t, aj the well-known theorem on division with remainder we obtain (25) a* = Ojtfo + fi, «i = + • • • ,r

m_i = 0 < r

m < r

m_i < ... < ri < aj and r

m = (a,-,aj) = d. Let A = ( 0/

1 ~

X

j ), then by (25) we obtain V aj xi J A _ f ajqo + T\ -Xj \ _ / 1 9o \ / n -(xj + q

0Xi)\ V aj J V

0 1 / \

aj xi ) (l) (ri x^ \ . Denoting by x - = — {xj -f qoX{) and by A\ = I I in similar way \ üj Xi J A! = 1 0 \ / ri x

(p 1 7 I r

2 x

l - qiX^P j