Counting Integer Solutions to Curves of the Form x^n-c-ky=0

[Klaus_Hoffmann]

Let be a open curve on R^2 so $$x^{n}-c-ky=0$$ where k,n and c are integers, are there any methods to calculate or at least know if the curve above will have integer roots (a,b) so a^{n}-c-kb=0 ?? or perhaps to calculate the number of solutions as a sum (involving floor function) over integers of expressions like

$$[(x^{n}-c)k^{-1}]$$

 

[Hurkyl]

Reduce modulo k.

 

[Klaus_Hoffmann]

thanks, but however i think that solving $$x^{n}=c mod(y)$$ is even harder

 

[Hurkyl]

I suggested you reduce modulo k, rather than modulo y.

Actually, what you wrote is trivially easy to find solutions for, but they won't help you solve the original equation.

 

[matt grime]

Let be a open curve on R^2 so $$x^{n}-c-ky=0$$ where k,n and c are integers,

This is neither an open set, nor are all curves of this form.

are there any methods to calculate or at least know if the curve above will have integer roots (a,b) so a^{n}-c-kb=0 ??

yes, trivially there will be plenty, i.e. infinitely many of integer point (I don't think you mean root, by the way), on the curveif c is an n'th root mod k, and none if not.

This is the kind of question that eljose would ask.