How?

N≡0,1,2,3,4,5,6,7(mod8)
N^3≡0,1,8,27,64,125,216,343
≡0,1,0,3,0,5,0,7≡0,1,3,5,7(mod8)
任取3數,令其和為S
S={0,3,9,15,21,1,3,5,7,5,7,9,11,13,17,1,7,13,19,4,6,8,9,11,15}
≡{0, 3, 1, 7, 5, 1, 3, 5, 7, 5, 7, 1, 3, 5, 1, 1, 7, 5, 3, 4, 6, 0, 1, 3, 7}(mod8)
因2沒出現,所以只要是8的倍數加2都不會是任3立方數的和
故本題得證

8*1+2=2^3+1^3+1^3

Try to consider 8N+7