素数倒数幻方
素数倒数幻方(prime reciprocal magic square)是指用素数倒数及其倍数的循环小数各位数组成的幻方。有些素数的倒数则可以形成对角线和也满足条件的幻方。
考虑在十进制下的1/7,其小数为循环小数1/7 = 0·142857142857142857...,若再考虑其倍数,会看到这六个数字的循环排列:
1/7 = 0·1 4 2 8 5 7... 2/7 = 0·2 8 5 7 1 4... 3/7 = 0·4 2 8 5 7 1... 4/7 = 0·5 7 1 4 2 8... 5/7 = 0·7 1 4 2 8 5... 6/7 = 0·8 5 7 1 4 2...
若用上述数字形成方阵,每一列的和是1+4+2+8+5+7,即为27,每一行的和也是27,若不考虑对角线,因此可以形成一个幻方:
1 4 2 8 5 7 2 8 5 7 1 4 4 2 8 5 7 1 5 7 1 4 2 8 7 1 4 2 8 5 8 5 7 1 4 2
不过其对角线不是27。
考虑1/19的倍数,下一行是上一行的二倍,而小数位数似乎右移一位:
01/19 = 0.052631578,947368421 02/19 = 0.1052631578,94736842 04/19 = 0.21052631578,9473684 08/19 = 0.421052631578,947368 16/19 = 0.8421052631578,94736
分子乘以2会让小数的位数右移一位:
在1/19形成的方阵中,其最大周期为18,每一行及每一列的和是81,而且对角线也是81,完全符合幻方的条件:
01/19 = 0·0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1... 02/19 = 0·1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2... 03/19 = 0·1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3... 04/19 = 0·2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4... 05/19 = 0·2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5... 06/19 = 0·3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6... 07/19 = 0·3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7... 08/19 = 0·4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8... 09/19 = 0·4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9... 10/19 = 0·5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0... 11/19 = 0·5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1... 12/19 = 0·6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2... 13/19 = 0·6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3... 14/19 = 0·7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4... 15/19 = 0·7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5... 16/19 = 0·8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6... 17/19 = 0·8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7... 18/19 = 0·9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8...
在各素数在不同进制下,也可能会有相同的现象,以下是列表,列出素数、进制以及幻方和 ((进制-1) 乘 (素数-1) / 2:
素数 | 进制 | 幻方和 |
---|---|---|
19 | 10 | 81 |
53 | 12 | 286 |
53 | 34 | 858 |
59 | 2 | 29 |
67 | 2 | 33 |
83 | 2 | 41 |
89 | 19 | 792 |
167 | 68 | 5,561 |
199 | 41 | 3,960 |
199 | 150 | 14,751 |
211 | 2 | 105 |
223 | 3 | 222 |
293 | 147 | 21,316 |
307 | 5 | 612 |
383 | 10 | 1,719 |
389 | 360 | 69,646 |
397 | 5 | 792 |
421 | 338 | 70,770 |
487 | 6 | 1,215 |
503 | 420 | 105,169 |
587 | 368 | 107,531 |
593 | 3 | 592 |
631 | 87 | 27,090 |
677 | 407 | 137,228 |
757 | 759 | 286,524 |
787 | 13 | 4,716 |
811 | 3 | 810 |
977 | 1,222 | 595,848 |
1,033 | 11 | 5,160 |
1,187 | 135 | 79,462 |
1,307 | 5 | 2,612 |
1,499 | 11 | 7,490 |
1,877 | 19 | 16,884 |
1,933 | 146 | 140,070 |
2,011 | 26 | 25,125 |
2,027 | 2 | 1,013 |
2,141 | 63 | 66,340 |
2,539 | 2 | 1,269 |
3,187 | 97 | 152,928 |
3,373 | 11 | 16,860 |
3,659 | 126 | 228,625 |
3,947 | 35 | 67,082 |
4,261 | 2 | 2,130 |
4,813 | 2 | 2,406 |
5,647 | 75 | 208,902 |
6,113 | 3 | 6,112 |
6,277 | 2 | 3,138 |
7,283 | 2 | 3,641 |
8,387 | 2 | 4,193 |
相关条目
参考资料
Rademacher, H. and Toeplitz, O. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 158–160, 1957.
Weisstein, Eric W. "Midy's Theorem." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/MidysTheorem.html (页面存档备份,存于互联网档案馆)