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Sunday, June 07, 2009

More on the Above "Proof"

The above "proof" adduces the square-free integers--a very interesting set. For all finite sets P of primes, Product P ordered by Sum P (and within each tranche by Size P) give a canonical ordering of the square-free integers:

2, 3, 5, 6, 7, 10, 15, 14, 21, 30, 11, 35, 42, 13, 22, 33, 70, 26, 105, 39, 55, 66, 17, 210, 65, 77, 78, 110, 19, 34, 165, 51, 91, 130, 154, 38, 195, 231, 330, 57, 85, 102, 182, 23, 273, 385, 390, 462, 95, 119, 143, 114, 170, 46, 255, 455, 546, 770, 69, 133, 190, 238, 286, 1155, 285, 357, 429, 510, 910, 115, 187, 138, 266, 1365, 29, 399, 595, 715, 570, 714, 858, ...

This uncomfortable order reflects the uncomfortable properties of primes in general, which is of course why I like it.

Like the primes themselves, the square-free integers may be isolated by the Sieve of Eratosthenes thus:

1) Write down the integers greater than 1.
2) Start with the first square greater than 1 (that's 4).
3) Strike it and all multiples.
4) Take the next square.
5) If it remains, strike it and all remaining multiples.
6) Goto (4).

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