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?

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I have no idea what your "ir-modular" question means. There has been work on $ax^n+by^n=cz^n$, and the methods used to settle the Fermat problem have met with success in some cases. See, for example, Henri Darmon, A fourteenth lecture on Fermat's last theorem, CRM Proc Lecture Notes 36, pp 103-115; also, Darmon and Merel, Winding quotients and some variants of Fermat's last theorem, J Reine Angew Math 490 (1997) 81-100. – Gerry Myerson Apr 20 '11 at 6:32

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.