- #1

e2m2a

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- TL;DR Summary
- Cannot find proof asserted by Wikipedia article on a generalized Diophantine equation

There is an entry in Wikipedia at this link: https://en.wikipedia.org/wiki/Pythagorean_triple

Under elementary properties of primitive Pythagorean triples, general properties,sixth bullet from the bottom of this section, there is this generalized Diophantine equation:

x^2p + y^2p = z^2

Where: p ≥ 2.

The article asserts there is no integer solution to this Diophantine equation for all values of p ≥ 2:

I have a number of questions about this. First, is this assertion true? Second, where can I find the proof for this? There was no citation for a proof. And third, If there is a proof, would it use the method of infinite descent for this generalized expression?

Under elementary properties of primitive Pythagorean triples, general properties,sixth bullet from the bottom of this section, there is this generalized Diophantine equation:

x^2p + y^2p = z^2

Where: p ≥ 2.

The article asserts there is no integer solution to this Diophantine equation for all values of p ≥ 2:

I have a number of questions about this. First, is this assertion true? Second, where can I find the proof for this? There was no citation for a proof. And third, If there is a proof, would it use the method of infinite descent for this generalized expression?