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$$3x^2 + 2y^4 = z^4$$

How do I solve this?? I would like to use so-called "elementary number theory", not abstract algebra (e.g. $\mathbb{Z} ( \sqrt d)$) or elliptic curves.

Note: I'm not asking what the solutions are, but rather how to find them.

My instincts are:

  • search the internet (I compared this equation with the ~280 here on MSE, and tried a variety of similar searches on uniquation.com ...)
  • search the 3 number theory books that I have
  • try to find solutions "by inspection" (possibly after reducing the order of the variables)
  • do some magic with modular arithmetic
  • use Alpern's solver - which seemed to indicate that there are no solutions (though I might have made an illegal substitution, so to speak)

I was able to identify $A = 6, B = 3, C = 6$ as solutions of $ \ 3A + 2B \ ^2 = C \ ^2$, but those aren't squares!

What is the number-theoretic approach to such problems? Is there a general method?

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    For your simplified problem, you should have kept the $x^2,$ with letter $U,V,W$ making $3 U^2 + 2 V^2 = W^2.$ This also has only the trivial solution $0,0,0.$ It is just quadratic residues, as in my answer below.2012-04-25

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