Az Egri Ho Si Minh Tanárképző Főiskola Tud. Közleményei. 1987. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 18/11)

Krystyna Bialek — Aleksander Grytczuk: The equation of Fermat in ...

- 81 KRYSTYNA BIALEK AND ALEKSANDER GRYTCZUK THE EQUATION OF FERMAT IN G

2Ck) AND QjVlT] 1. INTRODUCTION Let- G^Ck) be the set. of matrices of the form [ l ; ] where k is fixed integer such that k ^ 0 and are arbitrary integers. The purpose of this paper is to give a connection between the solution of Fermat equation in G^Ck

> and the solution of this equation in Q ^V' lc j . Some partial results concerning above problem are given in LI], 1

2 J

, Cd] (comp, tS

3). We prove the following theorems: THEOREM 1. The necessary and sufficient condition far tho equation C2> A

n + B

n = G

n , Cn ^ 2J) to have a solution in elements A,B,C <£ G^Ck} is the existence of the numbers a, ß, y e Q k j such that C3) a

n + ß

n = r

r' . THEOREM 2. Let K be a number field. If a,b,c e IC and a

2

m + b

2

m = c

2

m