All integer solutions​ of $x^4 + y^4 + z^4 - w^4= 1995$ This question is in the book 'The thrill and challenge of precollege mathematics'. I intend to attack this problem using Fermat's Little Theorem (FLT).
Notice that each term on LHS must be either of the form $5k$ or according to FLT $5k +1$ but the if $x^4$ is of the form $5k$ then $x=5c$ for some integer c. I can't think of any think after this.
I have also used the property that any fourth power when divided by $4$ leaves remainder $1$ if it is odd and $0$ if it is even. Using this I think there are only $2$ things possible. Either, $x,y,z$ are odd and $w$ is even or $x,y,z$ even and $w$ odd.
Thanks in advance. Don't give me answer tho just tell me which concept to apply, or if this is right approach, the how to attack the problem.

You can use a stronger result about fourth powers: $$x^4 \equiv 0,1 \mod 16$$
