prove the equation $x^4 + y^4 + z^4 =3009$ has no integer solutions.

Question

prove the equation $x^4 + y^4 + z^4 =3009$ has no integer solutions. I tried modulo 3 and find that it is possible that has integer solutions when $x,y,z$ are all congurent to 0 and 3009 is also congruent to 0 (mod3). And don't know where I get it wrong.

Answer 1

Modulo $5$, the fourth powers are $0$ and $1$ only. The sum of three of them is $0,1,2$ or $3$. But $3009\equiv4\pmod5$.