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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) 81-100.2011-04-20

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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.

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    @alan, you have a point, but then if you look at the question you'll see that $a^2=2b^2$ has that same solution. So quanta and I both should have been more careful to specify positive solutions, or nonzero solutions, or nontrivial solutions.2018-07-15