Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1993. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 21)

Aleksander Grelak: On The Equation (x2 - 1 )(y2 - 1) = z2

ALEKSANDER GREIAK ON THE EQUATION (x

2 -1)(/ -1) = z

2 ABSTRACT: In this paper we get an explicit form of the formulae for all solutions in integers x,y,z of the Diophantine equation (1) (x

2 -\)(y

2 - X) = z

2. The equation (1) has been consider by K. Szymiczek [2] for the case when x~a> 1 is a fixed integer. He proved that in this case the equation (1) has infinitely many solutions in integers x,y for every fixed integer a > 1. Let T

n(u) - cos (ft arccos u) be well-known Tchebyshev polynomial. In 1980 R. L. Graham [1] proved that all solutions of the equation (1) in integers x,y,z are given by the following formulae: (2) x=T

m(u), y = T

m(u), z = ^{T

n+

m(u)~ T

n_

m(uj). We note that the formulae (2) are effective but not easy to practical determination of the solutions of (1). In this paper we prove the following theorem: 91