# 142857: What a Fascinating Number

I've just been introduced to a rather extra-ordinary number: 142857.

Now as it stands, it doesn't look particularly fascinating or special but it does when you start to play with it. So lets get playing.

Lets start with some multiplication...

142857 x 1 = 142857 142857 x 2 = 285714 142857 x 3 = 428571 142857 x 4 = 571428 142857 x 5 = 714285 142857 x 6 = 857142

Oooooooo, well look at that. Do you see it?

Notice how each answer contains exactly the same individual numbers, just shifted slightly. It's easier to see if I list the lines as follows:

142857 x 1 = 142857 142857 x 3 = 428571 142857 x 2 = 285714 142857 x 6 = 857142 142857 x 4 = 571428 142857 x 5 = 714285

Coool.

Now what about multiplying it by 7?

142857 X 7 = 999999

Whoops!! Don't worry, it's not over. That's actually quite significant. Look what happens when you split the digits into pairs or triplets and add them up:

142 + 857 = 999 14 + 28 + 57 = 99

Hmmmm. Interesting. 9s. But what about multiples greater than 7? Well, there's a pattern there too...

142857 × 8 = 1142856 142857 × 9 = 1285713 142857 x 26 = 3714282 142857 x 142857 = 20408122449 142857 × 133 = 18999981

See it? Probably not, but you will when I tell you the secret: add the last six digits to the remaining digits and repeating this process until you have only six digits left...

1 + 142856 = 142857 1 + 285713 = 285714 3 + 714282 = 714285 20408 + 122449 = 142857 18 + 999981 = 999999

Whoops, that last one didn't work. Or did it? What's significant about 133? It's a multiple of 7. Hmmmm, that number 7 seems to be quite significant, and indeed it is... look what 1 ÷ 7 gives you...

1 ÷ 7 = 0 . 142857 142857 142857 142857 142857 142857 14...

Well look at that, a never ending list of our magic number.

So what's so special about 142857? Well, according to Wikipedia it's the best-known cyclic number in base 10 (the number system we use). If you multiply the number by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself.

Now wasn't that interesting? Now I've finished amazing you with my random findings, back to work.