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Tue, 28 Apr 2009

Factoring Factorials

Take a factorial. Any factorial. Factor it*. Notice a Pattern.

That pattern is that if you list all of the prime factors in order and the number of times they appear, at no point does any larger factor appear more often than any smaller factor. This would appear to be obvious, and it "obviously" applies to all factorials** (I've verified it by hand up to 28!, at which point I got bored). But I'm finding it hard to put a proof into words - or more concisely but equivalently - into symbols.

* this is easy. While factoring 1124000727777607680000 might be quite hard, if you know that it's 22! it becomes trivial because we know that it is divisible by 2, 3, 4, 5, ..., 21, 22, each of which is trivially factorable. Given that each number is the product of a unique set*** of factors, the factors of 22! can only be the set of all the factors of all the numbers we multipled to get 22!. Easy!

** it becomes obvious when you consider Eratosthenes' method for testing primality.

*** yes, I know, it's not really a set.

Posted at 17:53 by David Cantrell
keywords: geeky | maths
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