## Re:christmas challenge?

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• ... 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

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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