Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1989. 19/8. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 19)

K. Dialek — A . Grytzuk: Some remarks on certain diophontine equations

- 26 PROO

F. First, we prove that if Cx,y)=i, n<m and C3. 3) x

n + y

n j x

w - y

w, thr*n C3.d> n I m . Since m > n thus m =• nq + r , where 0^r<n. Ve consider two cases: Ci) q is an odd number, Cii) q is an even number. In the case Ci) , we have C3.5> x

m-y

r = x

r [x

n+y

n] [x

nc

qi

í-x

nt

<í2

,y+. ..-y"

c

q~

1

D]-y

n

q [x

r+y

r]. Since C3. 6) j •= 1 thus from C3. 3) and C3.5) we get x

n + y

n x

r + y

r , which is impossible if r>0. Thus r«0 and therefore nJm. In case Cii> we have q » 2

a q* , where C2,q

,)o<l , «->-! and therefore for m =» n 2

n q* + r , OSr<n we get C3. 7) Since x

M-y

m=x

r r, .-,2° r 02

a [(

x') - [

yn

q

} J thus by assumption C3.3) and (3.7) we obtain x

n + y

n I x

f • y

f , and similarly as above it is impossible if r>0. Thus r=0 and n|m follows. Since <x,y)=i , C3. 8) [x,x

n+y

n]=»l and [y,x

n+y

n ]«i