问题及解答

立方数指一个整数的三次方 $n^3$. 这里我们仅考虑非负整数.

下面是 $n=0,1,2,\ldots,999$ 的三次方(以十进制表示)

(0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 
3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 
19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 
54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97336, 103823, 110592, 
117649, 125000, 132651, 140608, 148877, 157464, 166375, 175616, 185193, 195112,
205379, 216000, 226981, 238328, 250047, 262144, 274625, 287496, 300763, 314432,
328509, 343000, 357911, 373248, 389017, 405224, 421875, 438976, 456533, 474552,
493039, 512000, 531441, 551368, 571787, 592704, 614125, 636056, 658503, 681472,
704969, 729000, 753571, 778688, 804357, 830584, 857375, 884736, 912673, 941192,
970299)

请问 $n^3$ 是否有某些特征? 并请证明.

1. 末尾数字以 0,1,8,7,4,5,6,3,2,9 循环出现. 这个很好理解, 只需考虑模 10

\[
(10m+k)^3\equiv k^3\pmod {10}
\]

Program:

软件: ARIBAS

代码:

function cubevec(len: integer): array;
var 
   vec: array[len];
    i: integer;
begin
    for i:=0 to len-1 do
        vec[i]:= i**3;
    end;
    return vec;
end.