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

-iL KRYSTYNA DIALEK AND ALEXANDER GRYTCZUK SOME REMARKS ON CERTAIN DIOP

HANTI

NE EQUATIONS ABSTRACT: The paper gives resutte on ihe nolutionn of ihe equation& a x

n+a

Jx

n~

1y+. . . +a y

n=m, x

p+ y

P

i= z

2 , O i 4 . 1 + 1 + 1 an

d 5 = 1 + 1+1 nxyz nxyz' where m

fn art? iniegers and p C>3) is a prime. 1. Introduciion In this paper we give Rome remarks concerning the following' problemene: 1° Let i>

n<x,y) denotes the form of degree n£3 and let m € Z then the equation Ci.l). x, y)*=m has s

o called regular solutions in <x,y> « Z

2. 2° If Cx, y)

1

1! and p>3 is a prime and the pijiml.joii CI

.2) x

p + y

p » z

2 has a solution in integers x,y,z then p j z or p j <pC z ) , where >p denotes the Euler function. 3° Let <1.3) 4=1+1+1 nxyz and CI. 4) 5-1+1 + 1 nxyz Erdős and Straus conjectured,, that the equation <i.3) has a positive solution in integers x.y,z for every natural number n £ 2 Csee [41, Open problems, p. 50 ; No IB). Similar conjecture was posed by Sierpinski Csee Í4J, Open