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Re:christmas challenge?

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  • Alan Eliasen
    ... For a measure of smoothness, I ll choose n-smoothness. This means that the number has no prime factor larger than n. This is a common criterion for
    Message 1 of 7 , Dec 29, 2006
      Paul Leyland wrote:
      > A slightly more difficult (but only slightly) pair of questions is:
      > which is/are the smoothest Soduko number(s) and which is/are the least
      > smooth? Choose any justifiable measure of smoothness you wish and, of
      > course, justify your choice.

      For a measure of smoothness, I'll choose n-smoothness. This means
      that the number has no prime factor larger than n. This is a common
      criterion for smoothness required for some factoring algorithms to
      succeed. (e.g. Pollard p-1.)

      Note that all of these numbers have factors of 9.

      In that case, the smoothest numbers are the 7-smooth:

      619573248 2^12 * 3^2 * 7^5
      948721536 2^7 * 3^2 * 7^7

      There are then 7 numbers which are 11-smooth,
      4 numbers which are 13-smooth,
      12 17-smooth,
      18 19-
      25 23-
      38 29-

      The least smooth number could be argued to be
      987654321 3^2 * 109739359

      Or, the next-least-smooth number that does not leave a prime residue
      when divided by 9 is:

      987651342 2 * 3^2 * 54869519

      A program in my own programming language "Frink" is here:
      http://futureboy.us/fsp/colorize.fsp?fileName=SudokuSmooth.frink

      Frink can be obtained here:
      http://futureboy.us/frinkdocs/

      --
      Alan Eliasen | "Furious activity is no substitute
      eliasen@... | for understanding."
      http://futureboy.us/ | --H.H. Williams
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