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

- 25 arid therefore m|a for j=0,l,...,n. Since d = ja^.a , j ^ o 1 . .., a

r j

, thus d ja

. and therefore rn j d . It contradicts to our conditions Jm|>i and mjd GO the proof is complete. We remark that by similar method we can prove the well-k^nown Lagrange's theorem concerning the number of solutions of the congruence fCx)^0 Cmod p), where f e Zixl and p is a prime Ccomp.t3]>. 3. On the equation x

p + y

p " z

2 In 1977 G.

Terjanian E81 proved thnt if the equation X

2

p 4- y

2

p - Z

2

p , where p is odd, has a solution in integers x,y,z , then 2p j x or 2pIy . In 1981 A.Rotkiewicz töl improved this result showing that if x

2

p + y

2

p « z

2

p has o solution in integers x,y,z, where p is and odd prime, then 8p

?jx or 8p

Jjy . In 1982 A.Rotkiewicz [61 obtained that if Cx,y)=l and p>3 is a prime and pjz and 2|z or p-fz and 2fz , then the equation C3. 1> x

p + y

p ® z

2 has no solution in integers x,y,z . For other results see also E

71. In this part we prove the following theorem: THEOREM 2, Suppose that Cx,y)=i and p>3 is a prime. If the equation (3.1) has a solution in integers x,y,z , then (3.2> p|z or pj^Cz) , where <p denotes the Euler function.