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Show that the Diophantine equation $x^{4} - y^{4} = z^{2}$ has no solutions in nonzero integers using the method of infinite descent.

Thanks for any help on this proof. Infinite descent has me kind of stumped because the book does not clearly explain the concept.

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Wikipedia article on infinite descent gives a proof of this. See also Proof of Fermat's Last Theorem for specific exponent. –  Martin Sleziak Apr 29 '12 at 7:52
The idea of infinite descent is that if you can prove that form a positive solution you can find a "smaller" positive solution, then this would set up a potential infinite sequence of smaller and smaller positive integer solutions. Since the positive integers don't have infinite strictly descending sequences, that means the sequence can never get started. –  Arturo Magidin Apr 29 '12 at 20:39

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