Without actual evaluation, prove that 2015 is not expressible as the sum of precisely three perfect squares.

As 2015 is worth 7, mod 8, it cannot be expressible as the sum of 3 squares, each of which must be congruent to {0,1,4}, mod 8.

As 2015 is odd, one or all 3 squares must be odd:

Odd + odd + odd= 1+1+1 = 3 mod 8.

Odd + even + even  = 1+0+0, 1+0+4, 1+4+4, mod 8; or {1,5} mod 8.

Posted by broll on 2015-07-08 12:04:49