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

Theorem: Let < A

X,B

X > denote the least positive solution of the Pell's equation A

2 - DB

2 -1. Then all solutions of the equation (1) in the integers x,y,z are given by the formulae x = ~ (AÍ+IJdbJ +(A

1-T/DBJ 1 1 z = — 4 [A

X+4DB

X)

] +{A

X-4DB

X)

J [A, + YFJDB\

)' - (A, - YFDB\

)' I {A, + 4DB

X Y - (A

X - JDB

X ) where ij are arbitrary positive integers. In the proof of our Theorem we use of the following Lemma. Lemma. Let < A

X,B

X> denote the least positive solution of the Pell's equation A

2 - DB

2 = 1 and let < A

i

, B

t >denote i-th solution of this equation. If the equation (1) has a solution in integers x,y then there exists a positive integer D such that for some ij we have x = A

i and y -A

y Moreover if for every squarefree D and every ij we take x = A

i and y = A

; where <A

i,B

j> and < Aj

, Bj > are the solutions of the equation A

2 - DB

2 = 1 then the numbers x,y satisfy the equation (1) with uniquely determined z. Proof. Suppose that integers x,y,z satisfy (1). Let (x

2 -l,y

2 -l) = d = Du

2, where D denotes the squarefree kernel of d

. Then we have 92