Prime Permutations

A few years ago, I played with fractions whose denominators are full reptend primes.  The fractions have decimal expansions that repeat, as do most fractions.  However, the repeating block for one fraction with a given denominator of this type is permutation of that for another fraction with the same denominator.  For example, consider the denominator 7.  The decimal expansions of its proper fractions are:
  1. 1/7 = 0.142857142857142857142857142857…
  2. 2/7 = 0.285714285714285714285714285714…
  3. 3/7 = 0.428571428571428571428571428571…
  4. 4/7 = 0.571428571428571428571428571428…
  5. 5/7 = 0.714285714285714285714285714285…
  6. 6/7 = 0.857142857142857142857142857142…
In each case, the repeating block is in bold.  Notice that the block for 1/7, 142857, is the same as it is for 2/7 (285714), except permuted by two spots.  That is, if you take the “14” from the 1/7 block off of the front end and put it on the back end, then you have the block for 2/7.  Likewise, the blocks for 3/7 and 4/7 are permutations of each other by three places—between the two fractions, the first half and second half of the blocks are swapped.  There are six different cyclic permutations of the block 142857, one for each fraction.

The same thing happens with other fractions whose denominators are full reptend primes.  For example, 19 is a full reptend prime.  When used as a denominator, its fractions repeat with a block of 18 digits.  Here are some of them:
  • 1/19 = 0.05263157894736842105263157894737…
  • 4/19 = 0.21052631578947368421052631578947…
  • 9/19 = 0.47368421052631578947368421052632…
Compared with the block for 1/19 (052631578947368421), that for 4/19 is shifted three places to the right, and that for 9/19 is shifted nine places, swapping the first and second halves of the 1/19 block.

The comparisons need not be only with the first fraction’s block; any block can be compared with any other block (for the same denominator) to find out how many places it has been shifted, relative to the reference block.  Let’s return to 7.  In the table below, each row is for one of the six proper fractions with a denominator of 7.  Each column is for the repeating block of one of the six fractions.  The cells, then, give the place in the decimal expansion (row) where the particular repeating block (column) shows up.  For example, the value in the cell for row 2/7 and column 1/7 is 5 (yellow highlight), meaning that 1/7’s repeating block (142857) appears beginning at the fifth place in 2/7’s decimal expansion.  Likewise, the 3 in the cell intersecting the 5/7 row and the 3/7 column (orange highlight) means that 3/7’s repeating block (428571) begins at the third place in 5/7’s decimal expansion.

Reference fraction 1/7
1/7 = 0.142857…132564
2/7 = 0.285714…516342
3/7 = 0.428571…621453
4/7 = 0.571428…354126
5/7 = 0.714285…243615
6/7 = 0.857142…465231

You may notice some patterns in the table.  The diagonal from the top left to the bottom right is all 1s.  This is because the reference fraction and the comparison block’s fraction are the same, so the comparison block always begins at the first place of its decimal expansion.  Also, the other diagonal, from the bottom left to the upper right, is all 4s.  In fact, all of the values in the table are arranged in diagonal bands, as this figure shows.  Each colored line shows a band for the same values, with each value being represented by a different color.  Extending the lines and removing the numbers and grid reveals the symmetry in the pattern.


Larger full reptend prime denominators have similar, but more detailed, patterns.  19 has 18 different proper fractions, as opposed to six for a denominator of 7.  Here is 19’s pattern, in black and white, to accentuate the structure, and in color to indicate the values.

The relatively sparse areas in the center and at the corners give a hint as to what happens with larger denominators.  Here are the patterns for denominators of 61 and 193 (193’s is off-center to highlight the fractal detail).

Finally, here is an image for a denominator of 1019, which is about the limit of my current system.