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
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