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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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Do there exist any cuboids, five of whose seven distinct inter-vertex lengths are integers a...

There don’t seem to be any small ones. Perfect cuboids would be examples, so a proof that no examples exist that are not perfect cuboids would also be very interesting.
Aubrey de Grey's user avatar