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Since $x^n+y^n=z^n$ has no solutions in integers for $n \geq 3$ I started to think about polynomials of degree $\leq n-1$ which need to be added to $x^n+y^n$ so that $x^n+y^n+p_n(x,y)=z^n$ has an infinite number of solutions.

Does such a polynomial always exists?

Of course, for different $n$´s the degrees of $p_n$´s might be different.

And , as a starting point to exclude trivialities , $x^n+y^n+p_n(x,y)$ should not be $n$-th power of a polynomial.

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  • $\begingroup$ You could consider the case z=x+y-c for a given positive integer c, if you are willing to relax the degree condition slightly. Gerhard "That Gives Infinitely Many Solutions" Paseman, 2020.03.10. $\endgroup$
    – Gerhard Paseman
    Mar 10, 2020 at 21:10
  • $\begingroup$ @GerhardPaseman But that substitution also changes the degree of the equation and I would like that not to happen. $\endgroup$
    – user153451
    Mar 10, 2020 at 21:14
  • $\begingroup$ I mean, only -c is then to the power of n. $\endgroup$
    – user153451
    Mar 10, 2020 at 21:18
  • $\begingroup$ You need to exclude some other trivial solutions, such as $-x^{3m}-y^{3m}+(x^3+q(x,y)+y^3)^m$, where $\deg q(x,y)\leq 2$. You could require that $n$ is prime or that $x^n+y^n+p_n(x,y)$ is not the $n$th power of a polynomial. $\endgroup$
    – Richard Stanley
    Mar 10, 2020 at 21:44
  • $\begingroup$ @RichardStanley That´s true, I would like to exclude all the trivial solutions, but I do not know how to characterize them with one or two conditions. $\endgroup$
    – user153451
    Mar 10, 2020 at 21:48

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