Prove that there does not exist integers $a,b,c,d$ such that $8a^4+4b^4+2c^4=d^4$.

Question

Prove that there does not exist integers $a,b,c,d$ such that $8a^4+4b^4+2c^4=d^4$.

Well-ordering may be useful.

Answer 1

Let $v_2(n)$ be the largest $k\in \mathbb N$ such that $2^k|n$

Notice that $v_2(8a^4),v_2(4b^4)$ and $v_2(2c^4)$ are all distinct, since they are distinct $\bmod 4$.

We conclude that $v_2(8a^4+4b^4+2c^2)$ is equal to $\min(v_2(8a^4),v_2(4b^2),v_2(2c^2))$. And this number is not a multiple of $4$.

On the other hand $v_2(d^4)$ is clearly a multiple of $2$.

Answer 2

Put another way: the function $$ f(a,b,c,d) = 8 a^4 + 4 b^4 + 2 c^4 - d^4 $$ is homogeneous. If there is any solution with at least one of the variables nonzero, we can divide out by any common factor and produce a solution with $$ \gcd(a,b,c,d) = 1. $$

The argument in the other answer shows that any answer is made up of four even variables, therefore the gcd is not one. This contradicts the assumption of existence of a solution with at least one variable nonzero. That is, the only solution is all $0.$