Chapter 4 - Elementary Number Theory and Methods of Proof - Exercise Set 4.1 - Page 161: 17

$2=1^{2}+1^{2}$ $4=2^{2}$ $6=2^{2}+1^{2}+1^{2}$ $8=2^{2}+2^{2}$ $10=3^{2}+1^{2}$ $12=2^{2}+2^{2}+2^{2}$ $14=3^{2}+2^{2}+1^{2}$ $16=4^{2}$ $18=4^{2}+1^{2}+1^{2}$ $20=4^{2}+2^{2}$ $22=3^{2}+3^{2}+2^{2}$ $24=4^{2}+2^{2}+2^{2}$

Although inelegant, the method of exhaustion is probably the simplest way to prove a conjecture such as this one. See example 4.1.5 for more on this method of proof.