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

- 27 and therefore from Euler's theorem we have <3. 9) and C 3.10) A* +

y ^ s i [mod[x

n+y

n]] y^ ~'

y ^ s i [modjx

n+y

n]]. From C3.9) and C3.10) we get *>(x"+y

n] _ <f [x

n+y

nJ .ri . I C3.ll) x +y By C3.3) , C3.14) it follows that C3.12) n j <p (x

n+y

nJ. If the equation C3.1) has a solution in integers x,y>x such that Cx,y)=i , then C3.Í3) v [x

p+y

pJ - v [xjFrom C

3.13) and C3.12) we get C 3. 14) p j p It is easy to see that <p fz

2J *= z -pCz) for arbitrary fixed integer z, tiierefore we get p I z or p j'/'Cz) and the proof is finished. &. On a conjecture of Erdős — Straus and Slerplnski In this part we prove two theorems. THEOREM 3

. The equation