# (Ln(*x*))^{3}

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3:36pm on Wednesday, 29th October, 2008:

## My Favourite Number

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When I was at primary school, my favourite number was 7. This was because: you couldn't roll it on one die but it was the commonest roll on two dice; it was the number of ranks in dominoes (most people thought there were 6, but forgot about blanks); the layout looked good on playing cards; it didn't have an easy-to-remember times table.

After finding out that 7 was regarded by many people as a lucky number, this put me off somewhat. I briefly flirted with 49 (7 squared), on the grounds that I could remember it when most of my schoolfriends couldn't, but that eventually wore off too. I went for several years without a favourite number.

Then, at age 11, I discovered 142,857.

Oh wow, what a number! Its digits are the repeating part of the fraction 1/7, or, put another way, it's a seventh of 999,999. It has some wonderful properties.

Basically, it's cyclical. (Dropping the commas to make it easier to see) 142857+142857=285714. Adding 142857 again gives 428571, 571428, 714285, 857142, then 999999. If you keep on adding 142857, it still works. Next comes 1142856, so take that first 1 and add it to the final 6 digits and you get 142857 again. Add another and you get 1285713, which using the same trick returns 285714. You can do this with arbitrary numbers: 142857*1234=176285538, so 176+285538=285714. It always returns one of the 6 cyclic permutations or 999999.

Also, it has nice things with squares. From the above, we might expect 142857*142857=20408122449, and 20408+122449=142857 (this makes it a Kaprekar number, by the way). However, we might not expect (857*857)-(142*142)=714285.

Not only that, it has pleasing connections with repeated 9s. 142857*7=999999, OK, we know that. But what about 142+857=999? Or 14+28+57=99?

Finally, its factors are 3, 11, 13 and 37. It's merely happy coincidence that the last two of these are leet, but there's something else buried in there. 11, 13 and 37 go into 142857 once each; 3 goes into it 3 times. So it's not 3*11*13*37=142857, it's 27*11*13*37=142857. Lo and behold, 1+4+2+8+5+7=27. So that makes it a Harshad number, too.

Ah, 142,857: what a number!

No, I don't use it in any of my passwords or credit card PINs...

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Copyright © 2008 Richard Bartle (richard@mud.co.uk).