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    I am stuck on this problem:

    Show that the equation x^2 + y^2 = z^5 + z has infinitely many solutions in positive integers x, y, and z having no common factor greater than 1.

    I have a proof; however it involves complex numbers and it uses some facts they probably don't expect you to know. (Although you could certainly use them if you did.) I'll give you them to you and see if you can figure out the rest. (It's not trivial even from here.)

    • There are infinitely many primes p such that p \equiv 1 (mod 4).
    • If a prime p is such that p \equiv 1 (mod 4), then p = a^2 +b^2 for some integers a,b.
    • c^2 +d^2 = (c+id)(c-id)

    There's probably some other way of doing it, but I hope this helps.
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Updated: November 13, 2017


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