Proof
(related to Corollary: Diophantine Equations of Congruences Have a Finite Number Of Solutions)
- Let $m > 1$ be a positive integer.
- If the Diophantine equation of congruences modulo $m$ $(f(x_1,\ldots,x_r))(m)=0(m)$ has no solution, then the number of solutions is $0 < \infty.$
- If it has a solution $(a_1(m),\ldots,a_r(m)),$ then, by congruences and division with quotient and remainder, there are at most $m$ ordered tuples solving this equation and containing the congruence class $a_k(m)$ at the index $k$, $1\le k\le r.$
- Since this argument is valid for all indices $k=1,\ldots,r$, there are at most $m^r < \infty$ ordererd tuples solving the equation.
- Altogether, the number of solutions is finite.
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