. . . . . "A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number. Consider a unit fraction, like 1/3 or 1/7. In base ten, the remainder, and so the digits, of 1/3 repeats at once: 0.3333...\u2009. However, the remainders of 1/7 repeat over six, or 7\u22121, digits: 1/7 = 0\u00B7142857142857142857... If you examine the multiples of 1/7, you can see that each is a cyclic permutation of these six digits: 1/7 = 0\u00B71 4 2 8 5 7...2/7 = 0\u00B72 8 5 7 1 4...3/7 = 0\u00B74 2 8 5 7 1...4/7 = 0\u00B75 7 1 4 2 8...5/7 = 0\u00B77 1 4 2 8 5...6/7 = 0\u00B78 5 7 1 4 2..."@en . "Un carr\u00E9 magique d'inverses de nombres premiers est un carr\u00E9 magique qui peut \u00EAtre obtenu en \u00E9crivant sur lignes successives le d\u00E9veloppement d\u00E9cimal des divisions de , ... . Le plus petit nombre \u00E0 avoir cette propri\u00E9t\u00E9 est le 19."@fr . . . . . . "1070325257"^^ . . "\u7D20\u6570\u5012\u6570\u5E7B\u65B9\uFF08prime reciprocal magic square\uFF09\u662F\u6307\u7528\u7D20\u6570\u5012\u6570\u53CA\u5176\u500D\u6578\u7684\u5FAA\u74B0\u5C0F\u6578\u5404\u4F4D\u6578\u7D44\u6210\u7684\u5E7B\u65B9\u3002\u6709\u4E9B\u7D20\u6570\u7684\u5012\u6570\u5247\u53EF\u4EE5\u5F62\u6210\u5C0D\u89D2\u7DDA\u548C\u4E5F\u6EFF\u8DB3\u689D\u4EF6\u7684\u5E7B\u65B9\u3002 \u8003\u616E\u5728\u5341\u9032\u5236\u4E0B\u76841/7\uFF0C\u5176\u5C0F\u6578\u70BA\u5FAA\u74B0\u5C0F\u65781/7 = 0\u00B7142857142857142857...\uFF0C\u82E5\u518D\u8003\u616E\u5176\u500D\u6578\uFF0C\u6703\u770B\u5230\u9019\u516D\u500B\u6578\u5B57\u7684\uFF1A 1/7 = 0\u00B71 4 2 8 5 7...2/7 = 0\u00B72 8 5 7 1 4...3/7 = 0\u00B74 2 8 5 7 1...4/7 = 0\u00B75 7 1 4 2 8...5/7 = 0\u00B77 1 4 2 8 5...6/7 = 0\u00B78 5 7 1 4 2... \u82E5\u7528\u4E0A\u8FF0\u6578\u5B57\u5F62\u6210\u65B9\u9663\uFF0C\u6BCF\u4E00\u5217\u7684\u548C\u662F1+4+2+8+5+7\uFF0C\u5373\u70BA27\uFF0C\u6BCF\u4E00\u884C\u7684\u548C\u4E5F\u662F27\uFF0C\u82E5\u4E0D\u8003\u616E\u5C0D\u89D2\u7DDA\uFF0C\u56E0\u6B64\u53EF\u4EE5\u5F62\u6210\u4E00\u500B\u5E7B\u65B9\uFF1A 1 4 2 8 5 72 8 5 7 1 44 2 8 5 7 15 7 1 4 2 87 1 4 2 8 58 5 7 1 4 2 \u4E0D\u904E\u5176\u5C0D\u89D2\u7DDA\u4E0D\u662F27\u3002 \u8003\u616E1/19\u7684\u500D\u6578\uFF0C\u4E0B\u4E00\u884C\u662F\u4E0A\u4E00\u884C\u7684\u4E8C\u500D\uFF0C\u800C\u5C0F\u6578\u4F4D\u6578\u4F3C\u4E4E\u53F3\u79FB\u4E00\u4F4D\uFF1A 01/19 = 0.052631578,94736842102/19 = 0.1052631578,9473684204/19 = 0.21052631578,947368408/19 = 0.421052631578,94736816/19 = 0.8421052631578,94736 \u5206\u5B50\u4E58\u4EE52\u6703\u8B93\u5C0F\u6578\u7684\u4F4D\u6578\u53F3\u79FB\u4E00\u4F4D\uFF1A \u57281/19\u5F62\u6210\u7684\u65B9\u9663\u4E2D\uFF0C\u5176\u6700\u5927\u9031\u671F\u70BA18\uFF0C\u6BCF\u4E00\u884C\u53CA\u6BCF\u4E00\u5217\u7684\u548C\u662F81\uFF0C\u800C\u4E14\u5C0D\u89D2\u7DDA\u4E5F\u662F81\uFF0C\u5B8C\u5168\u7B26\u5408\u5E7B\u65B9\u7684\u689D\u4EF6\uFF1A 01/19 = 0\u00B70 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1...02/19 = 0\u00B71 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2...03/19 = 0\u00B71 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3...04/19 = 0\u00B72 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4...05/19 = 0\u00B72 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5...06/19 = 0\u00B73 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6...07/19 = 0\u00B73 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7...08/19 = 0\u00B74 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8...09/19 = 0\u00B74 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9...10/19 = 0\u00B75 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0...11/19 = 0\u00B75 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1...12/19 = 0\u00B76 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2...13/19 = 0\u00B76 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3...14/19 = 0\u00B77 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4...15/19 = 0\u00B77 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5...16/19 = 0\u00B78 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6...17/19 = 0\u00B78 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7...18/19 = 0\u00B79 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8... \u5728\u5404\u7D20\u6578\u5728\u4E0D\u540C\u9032\u5236\u4E0B\uFF0C\u4E5F\u53EF\u80FD\u6703\u6709\u76F8\u540C\u7684\u73FE\u8C61\uFF0C\u4EE5\u4E0B\u662F\u5217\u8868\uFF0C\u5217\u51FA\u7D20\u6578\u3001\u9032\u5236\u4EE5\u53CA\u5E7B\u65B9\u548C\u3000\uFF08(\u9032\u5236-1) \u4E58 (\u7D20\u6578-1) / 2\uFF1A"@zh . . "Prime reciprocal magic square"@en . "Carr\u00E9 magique d'inverses de nombres premiers"@fr . "Un carr\u00E9 magique d'inverses de nombres premiers est un carr\u00E9 magique qui peut \u00EAtre obtenu en \u00E9crivant sur lignes successives le d\u00E9veloppement d\u00E9cimal des divisions de , ... . Le plus petit nombre \u00E0 avoir cette propri\u00E9t\u00E9 est le 19."@fr . "A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number. Consider a unit fraction, like 1/3 or 1/7. In base ten, the remainder, and so the digits, of 1/3 repeats at once: 0.3333...\u2009. However, the remainders of 1/7 repeat over six, or 7\u22121, digits: 1/7 = 0\u00B7142857142857142857... If you examine the multiples of 1/7, you can see that each is a cyclic permutation of these six digits: 1/7 = 0\u00B71 4 2 8 5 7...2/7 = 0\u00B72 8 5 7 1 4...3/7 = 0\u00B74 2 8 5 7 1...4/7 = 0\u00B75 7 1 4 2 8...5/7 = 0\u00B77 1 4 2 8 5...6/7 = 0\u00B78 5 7 1 4 2... If the digits are laid out as a square, each row will sum to 1+4+2+8+5+7, or 27, and only slightly less obvious that each column will also do so, and consequently we have a magic square: 1 4 2 8 5 72 8 5 7 1 44 2 8 5 7 15 7 1 4 2 87 1 4 2 8 58 5 7 1 4 2 However, neither diagonal sums to 27, but all other prime reciprocals in base ten with maximum period of p\u22121 produce squares in which all rows and columns sum to the same total. Other properties of prime reciprocals: Midy's theorem The repeating pattern of an even number of digits [7-1, 11-1, 13-1, 17-1, 19-1, 23-1, 29-1, 47-1, 59-1, 61-1, 73-1, 89-1, 97-1, 101-1, ...] in the quotients when broken in half are the nines-complement of each half: 1/7 = 0.142,857,142,857 ... +0.857,142 --------- 0.999,9991/11 = 0.09090,90909 ... +0.90909,09090 ----- 0.99999,999991/13 = 0.076,923 076,923 ... +0.923,076 --------- 0.999,9991/17 = 0.05882352,94117647 +0.94117647,05882352 ------------------- 0.99999999,999999991/19 = 0.052631578,947368421 ... +0.947368421,052631578 ---------------------- 0.999999999,999999999 Ekidhikena Purvena From: Bharati Krishna Tirtha's Vedic mathematics#By one more than the one before Concerning the number of decimal places shifted in the quotient per multiple of 1/19: 01/19 = 0.052631578,94736842102/19 = 0.1052631578,9473684204/19 = 0.21052631578,947368408/19 = 0.421052631578,94736816/19 = 0.8421052631578,94736 A factor of 2 in the numerator produces a shift of one decimal place to the right in the quotient. In the square from 1/19, with maximum period 18 and row-and-column total of 81, both diagonals also sum to 81, and this square is therefore fully magic:01/19 = 0\u00B70 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1...02/19 = 0\u00B71 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2...03/19 = 0\u00B71 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3...04/19 = 0\u00B72 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4...05/19 = 0\u00B72 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5...06/19 = 0\u00B73 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6...07/19 = 0\u00B73 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7...08/19 = 0\u00B74 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8...09/19 = 0\u00B74 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9...10/19 = 0\u00B75 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0...11/19 = 0\u00B75 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1...12/19 = 0\u00B76 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2...13/19 = 0\u00B76 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3...14/19 = 0\u00B77 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4...15/19 = 0\u00B77 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5...16/19 = 0\u00B78 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6...17/19 = 0\u00B78 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7...18/19 = 0\u00B79 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8... The same phenomenon occurs with other primes in other bases, and the following table lists some of them, giving the prime, base, and magic total (derived from the formula base\u22121 \u00D7 prime\u22121 / 2):"@en . "213581"^^ . . . . . . . . "5826"^^ . . . "\u7D20\u6570\u5012\u6570\u5E7B\u65B9\uFF08prime reciprocal magic square\uFF09\u662F\u6307\u7528\u7D20\u6570\u5012\u6570\u53CA\u5176\u500D\u6578\u7684\u5FAA\u74B0\u5C0F\u6578\u5404\u4F4D\u6578\u7D44\u6210\u7684\u5E7B\u65B9\u3002\u6709\u4E9B\u7D20\u6570\u7684\u5012\u6570\u5247\u53EF\u4EE5\u5F62\u6210\u5C0D\u89D2\u7DDA\u548C\u4E5F\u6EFF\u8DB3\u689D\u4EF6\u7684\u5E7B\u65B9\u3002 \u8003\u616E\u5728\u5341\u9032\u5236\u4E0B\u76841/7\uFF0C\u5176\u5C0F\u6578\u70BA\u5FAA\u74B0\u5C0F\u65781/7 = 0\u00B7142857142857142857...\uFF0C\u82E5\u518D\u8003\u616E\u5176\u500D\u6578\uFF0C\u6703\u770B\u5230\u9019\u516D\u500B\u6578\u5B57\u7684\uFF1A 1/7 = 0\u00B71 4 2 8 5 7...2/7 = 0\u00B72 8 5 7 1 4...3/7 = 0\u00B74 2 8 5 7 1...4/7 = 0\u00B75 7 1 4 2 8...5/7 = 0\u00B77 1 4 2 8 5...6/7 = 0\u00B78 5 7 1 4 2... \u82E5\u7528\u4E0A\u8FF0\u6578\u5B57\u5F62\u6210\u65B9\u9663\uFF0C\u6BCF\u4E00\u5217\u7684\u548C\u662F1+4+2+8+5+7\uFF0C\u5373\u70BA27\uFF0C\u6BCF\u4E00\u884C\u7684\u548C\u4E5F\u662F27\uFF0C\u82E5\u4E0D\u8003\u616E\u5C0D\u89D2\u7DDA\uFF0C\u56E0\u6B64\u53EF\u4EE5\u5F62\u6210\u4E00\u500B\u5E7B\u65B9\uFF1A 1 4 2 8 5 72 8 5 7 1 44 2 8 5 7 15 7 1 4 2 87 1 4 2 8 58 5 7 1 4 2 \u4E0D\u904E\u5176\u5C0D\u89D2\u7DDA\u4E0D\u662F27\u3002 \u8003\u616E1/19\u7684\u500D\u6578\uFF0C\u4E0B\u4E00\u884C\u662F\u4E0A\u4E00\u884C\u7684\u4E8C\u500D\uFF0C\u800C\u5C0F\u6578\u4F4D\u6578\u4F3C\u4E4E\u53F3\u79FB\u4E00\u4F4D\uFF1A \u5206\u5B50\u4E58\u4EE52\u6703\u8B93\u5C0F\u6578\u7684\u4F4D\u6578\u53F3\u79FB\u4E00\u4F4D\uFF1A \u57281/19\u5F62\u6210\u7684\u65B9\u9663\u4E2D\uFF0C\u5176\u6700\u5927\u9031\u671F\u70BA18\uFF0C\u6BCF\u4E00\u884C\u53CA\u6BCF\u4E00\u5217\u7684\u548C\u662F81\uFF0C\u800C\u4E14\u5C0D\u89D2\u7DDA\u4E5F\u662F81\uFF0C\u5B8C\u5168\u7B26\u5408\u5E7B\u65B9\u7684\u689D\u4EF6\uFF1A \u5728\u5404\u7D20\u6578\u5728\u4E0D\u540C\u9032\u5236\u4E0B\uFF0C\u4E5F\u53EF\u80FD\u6703\u6709\u76F8\u540C\u7684\u73FE\u8C61\uFF0C\u4EE5\u4E0B\u662F\u5217\u8868\uFF0C\u5217\u51FA\u7D20\u6578\u3001\u9032\u5236\u4EE5\u53CA\u5E7B\u65B9\u548C\u3000\uFF08(\u9032\u5236-1) \u4E58 (\u7D20\u6578-1) / 2\uFF1A"@zh . . . . . . . . . . . "\u7D20\u6570\u5012\u6570\u5E7B\u65B9"@zh . . .