MathDB

We know that

k

is a positive integer and the equation x^3+y^3-2y(x^2-xy+y^2)=k^2(x-y) (1) has one solution

(x_0,y_0)

with

x_0,y_0\in \mathbb{Z}-\{0\}

and

x_0\neq y_0

. Prove that i) the equation (1) has a finite number of solutions

(x,y)

with

x,y\in \mathbb{Z}

and

x\neq y

; ii) it is possible to find

11

addition different solutions

(X,Y)

of the equation (1) with

X,Y\in \mathbb{Z}-\{0\}

and

X\neq Y

where

X,Y

are functions of

x_0,y_0

3