It is a slick, and seemingly ad-hoc, technique often used to prove that a Diophantine equation has no solutions.

The equation $f(x_1,\ldots, x_k)=0$, with variables $x_i\in\mathbb{Z}$ and some elementary function $f$, is taken modulo some ingeniously chosen $n\in\mathbb{Z}^+$. Then by evaluating $f$ on the $k^n$ possible $k$-tuples $(x_1,\ldots,x_k)$ of residues modulo $n$, it is shown that $0$ is achieved in no case. Thus no solution exists in $\mathbb{Z}$ either.

A traditional classroom example is $x^2+y^2-3z^2=0$. Assume $\gcd(x,y)=1$ without loss of generality, and use modulo $3$.

The trouble is in coming up with the brilliant $n$. Has there been any work done on an objective way of finding $n$? Perhaps good heuristics, if nothing else? Restrict $f$ to some non-trivial case, say certain polynomials of given degree, if needed. Paper references would be nice.

It is a slick, and seemingly ad-hoc, technique often used to prove that a Diophantine equation has no solutions.

The equation $f(x_1,\ldots, x_k)=0$, with variables $x_i\in\mathbb{Z}$ and some elementary function $f$, is taken modulo some ingeniously chosen $n\in\mathbb{Z}^+$. Then by evaluating $f$ on the $k^n$ possible $k$-tuples $(x_1,\ldots,x_k)$ of residues modulo $n$, it is shown that $0$ is achieved in no case. Thus no solution exists in $\mathbb{Z}$ either.

A traditional classroom example is $x^2+y^2-3z^2=0$. Assume $\gcd(x,y)=1$ without loss of generality, and use modulo $3$.

The trouble is in coming up with the brilliant $n$. An example of a good heuristic principle is choosing modulo $20$ if there are powers of $4$ as, out of to $20$ possible residues, the only quartic residues are $0,1,5,16$. Are there similar mods for other powers?

In general, has there been any work done on an objective way of finding $n$? Perhaps good heuristics, if nothing else? Restrict $f$ to some non-trivial case, say certain polynomials of given degree, if needed. Paper references would be nice.