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

problems, p.

58, No 19) for the equation (1.4}. We give explicit form of solutionis of (1.3) for n=dk, 4k+2, dk+3, Bk+5j k=i

,2,

... and for n^dk, dk * 2 „ dk+3

; k=l

, 2, . . . of the equation (l.d). 2. Regulax' solutions of the aquatio

n <f> Cx, y) —

m

. J.H.Evertse t2J proved that if C

2.1} N « card ^ <x,y> « Z

2 j *>_(x,y)~m ^ , where nä3, ^(x,!) is a polynomial, which has three district roots and t=o>( |m|), where o(|m|) denotes the number of distinct prime divisors of|m| , then 2 (2. 2} N 5 n 15 ÜM + 6 * 7 (3) (t+i) An improvement of (2.2) was given by E. Flombieri and W.M.Schmidt [13. Lot FCx,y) « Ztx,yJ be an irreducible form of degree «»£3 and let N be the number of solutions of the t rr> equation (2.3) f

n(>

c»y} " m in integers x,y sue!» that (x,y)=l , then (2.4) N < C n n, m 1 where C is an absolute constant. If n > C . then 1 2 (2.5) where <x,y>= <~x,—y> . N < 213 R»

1*

1 n

, m In this part we consider so called regular solutions of (2.3). Let ip Cx, y) <s ZCx,yJ and n^3, and let (2.6) y (x, y) «= a x

n+a v

n"

iy+ ... +a y

n » m. N O 1 N