17 September 2007

142857

A web site devoted to the number 142857.

1/7 = 0.142857142857142857142857142857142857...

and, for example, 142857 * 2 = 285714 -- the same digits, but in a different order.

There are some facts I hadn't seen, too, like 8572 - 1422 = 714285, and the pair 1428572 = 20408122449, 20408+12249 = 142857. The first one has a nice "approximate" explanation, namely that (6/7)2 - (1/7)2 = 5/7 (multiply both sides by 106), but I don't even see why the second one has any chance of being true.
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3 comments:

  1. AnonymousSeptember 17, 2007 at 5:25 PM

    Well I think you just gave them a lot of traffic, I went to look and their bandwidth was exceeded.

    There was once a site for pi and e; I haven't look for them for a while.

    ReplyDelete
  2. Michael LugoSeptember 17, 2007 at 5:28 PM

    michael,

    I don't have that sort of effect. I learned about the site from reddit.com.

    ReplyDelete
  3. AnonymousSeptember 17, 2007 at 5:51 PM

    Look at the tails. divide to find:

    999999 = 20408 * 49 + 7

    and thus:

    (10^6 - 1)/49 = 20408 + 7/49 = 20408 + 1/7

    but we know:

    1/7 = 142857/(10^6 - 1)

    so we can calculate

    (10^6 - 1)/49 =
    (10^6 - 1) * 142857^2/(10^6 - 1)^2 =
    20408122449/(10^6-1)

    And now the geometric series says this is:

    20408122449 * \sum_{i=1}^\infty 10^{-6i} =
    20408.122449 +
    0.020408122449 +
    ...

    See it now?

    ReplyDelete