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How do I find the solubility of certain non-linear diophantine equations:

For eg.: $x^4+y^4=z^4$ is insoluble in $\mathbb{N}$, which is easy to prove by infinite descent, but $x^2+y^2=z^3$ has infinitely many solutions (which I am apparently stuck with).

The text from where this problem came (Elementary Number Theory, David M. Burton) contains a hint stating to choose $x=n(n^2-3) $ and $y=3n^2-1$ which trivializes the problem. I want to know how to understand the reasoning behind the choice of such substitutions.

Also, as a soft question, is it alright to simultaneously ready An Introduction to the Theory of Numbers by G.H. Hardy along with my present syllabus, or should I focus more on one text?

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The hint:

Let $x=y=2^n$ and $z=2^m$, where $(x,y,z)$ is a solution.

Thus, $$2^{2n+1}=2^{3m}$$ and prove that the equation $2n+1=3m$ has infinitely many natutal solutions.

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