What is your favorite number?

Most people, when asked this question, will answer with a single digit number like "7". Some folks will answer with a two digit number that has special meaning, like "13" or an anniversary date. I suppose there are a few who have a three digit favorite numbers, like the so-called devil's number "666". In general, most everyone keeps it pretty simple when it comes to favorite numbers.

I'm an astrophysicist, so I'm definitely not most everyone. That statement extends to favorite numbers as well. You see, I have a six digit favorite number. My favorite number is "142,857".

Really.

I became enamored of the number 142,857 in my childhood, while reading a book on mathematical diversions. I was the type who read such books in my spare time. I just loved solving puzzles (still do), and when one could combine some sort of story and an interesting mathematical viewpoint with a puzzle, all the better.

In the US, books like this were made popular by Martin Gardner. He wrote a column for Scientific American about recreational mathematics with a playfully intellectual bent. Wikipedia says he has over 70 books to his credit. Check them out if you like such things.

However, the book I was reading was not by Martin Gardner. I'm sure it was a British publication. I remember that it had less of a playful tone, and more of an expectation of interest. The material was presented in a more straightforward manner, with smaller hints of enthusiasm and whimsy than what American culture would produce.

The Ring Number

The number 142,857 was presented, somewhat matter of factly, as "the ring number". That title was simply defined by example.

Consider this: 2 times 142,857 equals 285,714. Do you notice what is special about that multiplication? Take a second to look carefully before reading on.

What is interesting about this multiplication is that the multiple of 142,857 contains the same digits. Further, it contains the same digits in the same order, if you allow for cyclical rotation. Simply move the "14" from the front end of the number to the back end, and you get 285,714.

OK. That's curious, but not overwhelming. Wait, we're just getting started.

Let's try 3 times 142,857. You should really do this. Bring up the calculator on your computer screen and punch in the numbers. Or, egads, work through the math yourself. That's what I had to do back in the stone ages when calculators and computers were not so prevalent. Go ahead, I'll wait.

The correct answer is 428, 571. Immediately you should see that this is also a cyclic rotation of the original number. Just take the "1" from the front end and put it on the back end. Now you get the idea of what makes a ring number. The multiples of 142,857 use the same digits, but in cyclic rotations.

Since there are six digits, the multiples from two times up through six times will each start with a different digit. You can put away your calculator, as It is not hard to figure out the other multiples. If you start with the next higher digit each time and  create a cyclic rotation, there you have it.

1 * 142,857 = 142,857

2 * 142,857 = 285,714

3 * 142,857 = 428,571

4 * 142,857 = 571,428

5 * 142,857 = 714,285

6 * 142,857 = 857,142

It is still good to check these answers by entering them into your calculator. Most folks will find these results hard to believe without some verification.

This marvelous quality is what makes 142,857 my favorite number. The crazy mathematical coincidence that you can start with any digit in the sequence, shift the numbers around, and you end up with a multiple of the original number. There is no other number quite like 142,857.

How do I know? Well, I investigated the general idea of ring numbers a long time ago.

Other Ring Numbers?

The book I read presented 142,857 as "the" ring number. However, it also showed a hint as to how one might generate other ring numbers. Simply take things one step further and multiply by seven.

7 * 142,857 = 999,999

Cool, huh? The ring number is exactly one-seventh of 999,999. It is almost one-seventh of a million. That fact led me to recognize that this string of numbers is the repeating part of the fraction one-seventh.

1 / 7 = 0.142857142857142857...

It also led me to speculate as to whether other fractions made from one over a prime number would form ring numbers. The next one in the series would be one-eleventh:

1 / 11 = 0.090909...

That fraction has only a two digit repeating sequence of "09". Still, multiply it by ten and you get "0.909090...", which shows that it does have ring number characteristics, although not very impressive.

The situation is better for one-thirteenth, which has a six digit repeating sequence:

1 / 13 = 0.076923...

3 / 13 = 0.230769...

4 / 13 = 0.307692...

9 / 13 = 0.692307...

10 / 13 = 0.769230...

12 / 13 = 0.923076...

Pretty cool, but half of the multiples are missing from the ring sequence. I find it not quite as satisfying.

I played with lots of ring numbers for a while, but never found one that was complete like 142,857. In particular, notice that the repeating part of one-thirteenth begins with a zero. To create the original ring number discussion, you would have to write it as "076,923". Then, it seems a bit contrived. Since all fractions less than 1/10 will begin with a zero, going to higher prime numbers will always produce this objection.

My simple conjecture that the repeating part of a reciprocal of a prime number would be a ring number turned out to be true. However, I could not find a way to generate all possible ring numbers. Perhaps there are others that can be found. As I said before, I did this work by hand, so I didn't write some computer code to brute force check all possibilities. That would be pretty easy to do today. If someone does it, send me the results.

I'm satisfied that there is just no ring number like 142,857. And that's why it is my favorite number.