This might not be solvable, but my NT teacher wanted us to give it a try.
Solve $x_1^4 + x_2^4+\cdots+x_7^4 = 1,000,007$.
What has been figured out so far:
Mod $8$ to get the equation $= 7 \pmod 8$. Also with $\bmod 8$, all the numbers must be odd as $0-0$. $1-1$, $2-0$, $3-1$, etc.
If you make use of $\bmod 5$, also, $5$ of the integers are divisible by $5$.