1/29_{[b=2]} = 0·0000100011010011110111001011… (l=28)

1/29_{[b=3]} = 0·0002210102011122200121202111… (l=28)

1/29_{[b=5]} = 0·00412334403211… (l=14)

1/29_{[b=7]} = 0·0145536… (l=7)

1/29_{[b=11]} = 0·04199534608387[10]69115764[10]2723… (l=28)

1/29_{[b=13]} = 0·05[10]9[11]28[12]7231[10]4… (l=14)

1/29_{[b=17]} = 0·09[16]7… (l=4)

1/29_{[b=19]} = 0·0[12]89[15][13][14]7[16]73[17][13]1[18]6[10]9354[11]2[11][15]15[17]… (l=28)

1/29_{[b=23]} = 0·0[18]5[12][15][19][19]… (l=7)

1/29_{[b=29]} = 0·1 (l=1)

1/29_{[b=31]} = 0·1248[17]36[12][25][20]9[19]7[14][29][28][26][22][13][27][24][18]5[10][21][11][23][16]… (l=28)

1/29_{[b=37]} = 0·1[10]7[24]8[34][16][21][25][19]53[30][22][35][26][29][12][28]2[20][15][11][17][31][33]6[14]… (l=28)

1/29_{[b=41]} = 0·1[16][39][24]… (l=4)

1/29_{[b=43]} = 0·1[20][32][26][29][28]7[17][34]4[19][11][37]2[41][22][10][16][13][14][35][25]8[38][23][31]5[40]… (l=28)

1/29_{[b=47]} = 0·1[29]84[40][24][14][27][25][43][35][30][37][12][45][17][38][42]6[22][32][19][21]3[11][16]9[34]… (l=28)

1/29_{[b=53]} = 0·1[43][45][36][29][12][42]… (l=7)

1/29_{[b=59]} = 0·2… (l=1)

1/29_{[b=61]} = 0·26[18][56][48][23]8[25][14][44][10][31][33][39][58][54][42]4[12][37][52][35][46][16][50][29][27][21]… (l=28)

1/29_{[b=67]} = 0·2[20][53]9[16][11][36][64][46][13][57][50][55][30]… (l=14)

1/29_{[b=71]} = 0·2[31][58][53][61][14][48][68][39][12][17]9[56][22]… (l=14)

1/29_{[b=73]} = 0·2[37][55][27][50][25][12][42][57][65][32][52][62][67][70][35][17][45][22][47][60][30][15]7[40][20][10]5… (l=28)

1/29_{[b=79]} = 0·2[57][16][27][19]5[35][32][54][38][10][70][65][29][76][21][62][51][59][73][43][46][24][40][68]8[13][49]… (l=28)

1/29_{[b=83]} = 0·2[71][45][65][68][57][20]… (l=7)

1/29_{[b=89]} = 0·36[12][24][49]9[18][36][73][58][27][55][21][42][85][82][76][64][39][79][70][52][15][30][61][33][67][46]… (l=28)

1/29_{[b=97]} = 0·3[33][43][46][80][26][73][56][83][60][20]6[66][86][93][63][53][50][16][70][23][40][13][36][76][90][30][10]… (l=28)

# Tag Archives: reciprocal

# More Narcissisum

The number 23 is special, *inter alia*, because it’s prime, divisible by only itself and 1. It’s also special because its reciprocal has maximum period. That is, the digits of 1/23 come in repeated blocks of 22, like this:

1/23 = 0·__0__434782608695652173913 __0__434782608695652173913 __0__434782608695652173913…

But 1/23 fails to be special in another way: you can’t sum its digits and get 23:

0 + 4 + 3 + 4 + 7 = 18

0 + 4 + 3 + 4 + 7 + 8 = 26

0 + 4 + 3 + 4 + 7 + 8 + 2 + 6 + 0 + 8 + 6 + 9 + 5 + 6 + 5 + 2 + 1 + 7 + 3 + 9 + 1 + 3 = 99

1/7 is different:

1/7 = 0·142857… → 1 + 4 + 2 = 7

This means that 7 is narcissistic: it reflects itself by manipulation of the digits of 1/7. But that’s in base ten. If you try base eight, 23 becomes narcissistic too (note that 23 = 2 x 8 + 7, so 23 in base eight is 27):

1/27 = 0·02620544131… → 0 + 2 + 6 + 2 + 0 + 5 + 4 + 4 = 27 (base=8)

Here are more narcissistic reciprocals in base ten:

1/3 = 0·3… → 3 = 3

1/7 = 0·142857… → 1 + 4 + 2 = 7

1/8 = 0·125 → 1 + 2 + 5 = 8

1/13 = 0·076923… → 0 + 7 + 6 = 13

1/14 = 0·0714285… → 0 + 7 + 1 + 4 + 2 = 14

1/34 = 0·02941176470588235… → 0 + 2 + 9 + 4 + 1 + 1 + 7 + 6 + 4 = 34

1/43 = 0·023255813953488372093… → 0 + 2 + 3 + 2 + 5 + 5 + 8 + 1 + 3 + 9 + 5 = 43

1/49 = 0·020408163265306122448979591836734693877551… → 0 + 2 + 0 + 4 + 0 + 8 + 1 + 6 + 3 + 2 + 6 + 5 + 3 + 0 + 6 + 1 + 2 = 49

1/51 = 0·0196078431372549… → 0 + 1 + 9 + 6 + 0 + 7 + 8 + 4 + 3 + 1 + 3 + 7 + 2 = 51

1/76 = 0·01315789473684210526… → 0 + 1 + 3 + 1 + 5 + 7 + 8 + 9 + 4 + 7 + 3 + 6 + 8 + 4 + 2 + 1 + 0 + 5 + 2 = 76

1/83 = 0·01204819277108433734939759036144578313253… → 0 + 1 + 2 + 0 + 4 + 8 + 1 + 9 + 2 + 7 + 7 + 1 + 0 + 8 + 4 + 3 + 3 + 7 + 3 + 4 + 9 = 83

1/92 = 0·010869565217391304347826… → 0 + 1 + 0 + 8 + 6 + 9 + 5 + 6 + 5 + 2 + 1 + 7 + 3 + 9 + 1 + 3 + 0 + 4 + 3 + 4 + 7 + 8 = 92

1/94 = 0·01063829787234042553191489361702127659574468085… → 0 + 1 + 0 + 6 + 3 + 8 + 2 + 9 + 7 + 8 + 7 + 2 + 3 + 4 + 0 + 4 + 2 + 5 + 5 + 3 + 1 + 9 + 1 + 4 = 94

1/98 = 0·0102040816326530612244897959183673469387755… → 0 + 1 + 0 + 2 + 0 + 4 + 0 + 8 + 1 + 6 + 3 + 2 + 6 + 5 + 3 + 0 + 6 + 1 + 2 + 2 + 4 + 4 + 8 + 9 + 7 + 9 + 5 = 98

Previously pre-posted (please peruse):

• Digital Disfunction

• The Hill to Power

• Narcissarithmetic #1

• Narcissarithmetic #2

# Three Is The Key

If *The Roses of Heliogabalus* (1888) is any guide, Sir Lawrence Alma-Tadema (1836-1912) thought that 222 is a special number. But his painting doesn’t exhaust its secrets. To get to another curiosity of 222, start with 142857. As David Wells puts it in his *Penguin Dictionary of Curious and Interesting Numbers *(1986), 142857 is a “number beloved of all recreational mathematicians”. He then describes some of its properties, including this:

142857 x 1 = 142857

142857 x 2 = 285714

142857 x 3 = 428571

142857 x 4 = 571428

142857 x 5 = 714285

142857 x 6 = 857142

The multiples are cyclic permutations: the order of the six numbers doesn’t change, only their starting point. Because each row contains the same numbers, it sums to the same total: 1 + 4 + 2 + 8 + 5 + 7 = 27. And because each row begins with a different number, each column contains the same six numbers and also sums to 27, like this:

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

= = = = = =

2 2 2 2 2 2

7 7 7 7 7 7

If the diagonals of the square also summed to the same total, the multiples of 142857 would create a full magic square. But the diagonals don’t have the same total: the left-right diagonal sums to 31 and the right-left to 23 (note that 31 + 23 = 54 = 27 x 2).

But where does 142857 come from? It’s actually the first six digits of the reciprocal of 7, i.e. 1/7 = 0·142857… Those six numbers repeat for ever, because 1/7 is a prime reciprocal with maximum period: when you calculate 1/7, all integers below 7 are represented in the remainders. The square of multiples above is simply another way of representing this:

1/7 = 0·142857…

2/7 = 0·285714…

3/7 = 0·428571…

4/7 = 0·571428…

5/7 = 0·714285…

6/7 = 0·857142…

7/7 = 0·999999…

The prime reciprocals 1/17 and 1/19 also have maximum period, so the squares created by their multiples have the same property: each row and each column sums to the same total, 72 and 81, respectively. But the 1/19 square has an additional property: both diagonals sum to 81, so it is fully magic:

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__…

First line = 0 + 5 + 2 + 6 + 3 + 1 + 5 + 7 + 8 + 9 + 4 + 7 + 3 + 6 + 8 + 4 + 2 + 1 = 81

Left-right diagonal = 0 + 0 + 7 + 5 + 5 + 9 + 0 + 3 + 0 + 4 + 2 + 8 + 7 + 5 + 6 + 7 + 5 + 8 = 81

Right-left diagonal = 9 + 9 + 2 + 4 + 4 + 0 + 9 + 6 + 9 + 5 + 7 + 1 + 2 + 4 + 3 + 2 + 4 + 1 = 81

In base 10, this doesn’t happen again until the 1/383 square, whose magic total is 1719 (= 383-1 x 10-1 / 2). But recreational maths isn’t restricted to base 10 and lots more magic squares are created by lots more primes in lots more bases. The prime 223 in base 3 is one of them. Here the first line is

1/223 = 1/22021_{3} = 0·

0000100210210102121211101202221112202

2110211112001012200122102202002122220

2110110201020210001211000222011010010

2222122012012120101011121020001110020

0112011110221210022100120020220100002

0112112021202012221011222000211212212…

The digits sum to 222, so 222 is the magic total for all rows and columns of the 1/223 square. It is also the total for both diagonals, so the square is fully magic. I doubt that Alma-Tadema knew this, because he lived before computers made calculations like that fast and easy. But he was probably a Freemason and, if so, would have been pleased to learn that 222 had a link with squares.