What’s So Special About 142857?

To be a Mathematician is to continue to ask the question of ‘why?’. Long after a normal person would be satisfied.

One lesson towards the end of primary school was taken by our headteacher. In it he decided to teach us all about repeating decimals, he did this using the decimal representations of the seventh fractions. From what I remember the lesson consisted of our headteacher writing out various seventh fractions as decimals and getting us to try and spot the pattern. The details of the rest of the lesson are hazy but I do remember that I was quite slow at spotting the pattern. What we were being challenged to notice was the fact that the decimal notation of the sevenths are made up of the same numbers in the same order starting in different places, like so:

1/7 = 0.14285714285714285714…

2/7 = 0.28571428571428571428…

3/7 = 0.42857142857142857142…

4/7 = 0.57142857142857142857…

5/7 = 0.71428571428571428571…

6/7 = 0.85714285714285714285…

7/7 = 0.99999999999999999999… (=1)

You can see that the number 142857 repeats in each of these fractions. I’m pretty sure the lesson ended with something akin to an ‘isn’t that cool?’ from the headteacher and much bemusement from the class. I hadn’t thought about this lesson until recently, and when I remembered the pattern I immediately thought ‘but why?’. An ‘isn’t that cool?’ explanation no longer satisfies me, and so I went of to find out what was really going on.

Luckily it didn’t take much googling to find out that 142857 is an example of a ‘cyclic number‘. Put simply a cyclic number is one whose digits permute as shown above when multiplied. Cyclic numbers can shown to be of the form:

Picture1

Where refers to the ‘number base’ which corresponds to the number of digits used. We use 10 digits (0-9) in the usual decimal notation (The binary numeral system which uses only 0’s and 1’s has a number base of 2 for example). Here p refers to a prime number. If we put b = 10 and p = 7 into the formula, we obtain the number 142857.

You may think that we could just follow this formula for every prime number to obtain an infinite amount of cyclic numbers, but unfortunately not every prime number provides us with a cyclic number. The first few values of p that provide us with cyclic numbers are as follows:

7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, …

But why these primes, what’s so special about them? Well it turns out that these are not just random primes, but these are the sequence of primes p such that 10 is a ‘primitive root modulo p‘. To explain that sequence is beyond the scope of this post, but it does show that mathematicians are continually asking the question of ‘why?’. We’ve briefly dove down the rabbit hole of the questions the number 142857 rises, so just keep in mind that Mathematicians around the world are partaking in this kind of reasoning across all areas of Mathematics, all the time.

I’m pretty sure my headteacher couldn’t have answered the question of ‘why?’ and even if they could have, the answer would have been far too advanced for our primary school minds to even comprehend. Still, being exposed to this phenomenon inspired me to research it myself, and isn’t that the ultimate goal of education?

EDIT: This video was recommended to me, it’s great and explains loads more properties of cyclic numbers I didn’t get around to!

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