sum of 4 squares

**Sally_Math** · April 15th 2009, 08:36 AM

If $8l(x^2+y^2+z^2+w^2)$ , show that $x,y,z,w$ are even.
I know I have to classify squares mod 8, but can someone show me how.

**ThePerfectHacker** · April 15th 2009, 09:26 AM

Originally Posted by Sally_Math

If $8l(x^2+y^2+z^2+w^2)$ , show that $x,y,z,w$ are even.
I know I have to classify squares mod 8, but can someone show me how.

If $a$ is even then $a=4k+1$ or $a=4k+3$ . Notice $(4k+1)^2 = 16k^2 + 8k + 1 = 8(2k^2+k) + 1$ . Also $(4k+3)^2 = 16k^2 + 24k + 9 = 8(2k^2+3k+1)+1$ . Therefore, an odd integer modulo 8 is congruent to 1. Therefore, $x^2+y^2+z^2+w^2 \equiv 1,2,3,4 (\bmod 8)$ if at least one of them is odd. Thus, we require all four to be even.

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