Our math teacher tells us that he read that showing Riemann hypothesis is true was equivalent to showing a particular diophantine equation doesn't have any solution. He asked us to make a list of cases in mathematics of the form:
Case A is true if and only if a particular diophantine equation doesn't have any solution.
My google searching return only the Riemann hypothesis.
For example, $n=1$ is not a congruent number if and only if $X^4+Y^4=Z^4$ has no nontrivial integer solution.
Reference: $1$ is not congruent because of Fermat's Last Theorem?
Another example is, that every elliptic curve of $\Bbb Q$ is modular if and only if $X^n+Y^n=Z^n$ has no nontrivuial integer solution for all $n\ge 3$ (Modularity conjecture, Taniyama-Shimura Conjecture and FLT).