Diophantine impossibility and irrationality (or similar) The Diophantine equation $$a^2 = 2 b^2$$ having no solutions is the same as $\sqrt{2}$ being irrational.
Are there any Diophantine equations which are related to the irrationality of a number that is not algebraic?

For a similar question with broader scope, the Diophantine equation $$x^n + y^n = z^n$$ implies a certain elliptic curves is "ir"-modular.
Are there more examples of this phenomenon?

The Diophantine equation $2^x=3^y$ having no solutions is the same as $\log3/\log2$ being irrational. It is known that $\log3/\log2$ is not algebraic. 
