Are there natural numbers $x$, $y$, $z$, such that $x^4+y^4-2^{3z}=3$?

Question

Are there natural numbers $x$, $y$, $z$, such that $x^4+y^4-2^{3z}=3$? Justify!!

I literally have no idea for this problem. I thought of doing the last digit but it doesnt help at all. I wonder if we can write $x^4+y^4$ as a product of two terms of something like this. I think using modulo can help solving this problem, but I don't get to any good idea.

Please help me!!

Answer 1

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Try looking at the equation modulo $8$.

Notice that $2^{3z}$ is divisible by $8$, so you would get that the sum of two fourth powers gives the residue $3$ mod $8$.