19 61

If you’ve played with fractions and decimals (and who hasn’t?), then you probably know that 7 is a special number.  The reciprocal of 7, 1/7, has a decimal expansion that repeats.  That’s not special—most fractions do.  However, what makes 7 (or 1/7) more special is that the block of repeating digits is as long as is possible (one less than the denominator--6 digits, in this case).  Also, the fractions 2/7 through 6/7 have decimal expansions that repeat with the same block (bolded below):
  • 1/7 = 0.142857142857142857142857142857…
  • 2/7 = 0.285714285714285714285714285714…
  • 3/7 = 0.428571428571428571428571428571…
  • 4/7 = 0.571428571428571428571428571428…
  • 5/7 = 0.714285714285714285714285714285…
  • 6/7 = 0.857142857142857142857142857142…
Alternatively, we can look at the repeating blocks as different, but as cyclic permutations of the block for 1/7.  That is, the block for 1/7 is 142857.  For 2/7, it is 285714, which is just 142857, shifted by two spots.  For 3/7, 428571 is shifted one spot from the original.  Likewise for the other three fractions.  This makes 7 a cyclic prime.  It’s not the only one; there are lots.  For a list, see entry A001913 in the Online Encyclopedia of Integer Sequences.

It turns out that 19 and 61 are also cyclic primes.  Just as 7 has six numbers in its repeating block, 19 has 18 and 61 has 60.  The blocks for 1/19 and 1/61 are:

·         1/19 = 052631578947368421…
·         1/61 = 016393442622950819672131147540983606557377049180327868852459…

Since I was born in 1961, I thought I’d try to do something with that.  It seemed reasonable to try to fit all of the fractions into an 18 x 60 box, those with a denominator of 19 being columns 18 rows long, and those with a denominator of 61 being rows 60 columns long.  For example, the blocks for fractions m/7 can fit into a 6 x 6 square; here are four of them:

3/7

6/7




4

8




2

5


2/7
2
8
5
7
1
4


5

1


4/7
5
7
1
4
2
8


1

2



Alas, for 19 and 61, it’s not quite that simple.  There are only 18 different decimal expansions for fractions of the form m/19, not 60, and there are 60 decimal expansions for fractions of the form m/61, not just 18.  Also, the actual numbers don’t generally line up, like they do for m/7.  But, I was able to find one case in which 21 blocks of repeating decimals fit, and another in which 24 blocks fit.  Here’s the upper left corner of the 21-block box:






14/19
11/19

2/19







7
5

1
37/61
6
0
6
5
5
7
3
7
7
0
49/61
8
0
3
2
7
8
6
8
8
5







8
9

2
1/61
0
1
6
3
9
3
4
4
2
6

This shows that the decimal expansion of 14/19 begins 0.73684, and that of 37/61 begins 0.6065573770.  Note that both share the 3 that is in the second row, seventh column.  One trick in setting up this box was ensuring that, whenever a row fraction and a column fraction share the same cell, that numeral is correct in both expansions.

Another challenge was how to present this an in aesthetically interesting fashion.  I came up with a solution, but will leave it to you to decide how interesting it is.  Rather than just writing the numerals, I used modified rose curves to represent them.  In the 21-block case, they are all shown using this alphabet, which shows 0, 1, 2, 4, and 8:


In the 24-block case, I tweaked the design a bit.  Also, I decided to render those numbers in the intersection cells (like 3, above, which is shared by 14/19 and 37/61) in negative.  Here are the symbols for 3, 5, 6, 7, and 9:


Now that you know what the symbols mean, you can read the image and there’s no need for labels, right?  So, without further ado, I present “19 61 a” and “19 61 b.”
19 61 a

19 61 b