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You are not logged in. #1 20090330 10:36:21
Number theoryThis is something I’ve just read about in H.E. Rose’s A Course in Number Theory. The proof is remarkably simple. Last edited by JaneFairfax (20090330 10:36:46) #2 20090330 12:01:35
Re: Number theoryThe proof relies on the multiplicative properties of the sigma function above. #3 20090401 05:02:39
Re: Number theoryWe have this interesting little result: The proof is only a few lines long. Also: Last edited by JaneFairfax (20090401 09:37:26) #4 20090401 09:31:32
Re: Number theoryThat makes it easy to compute the phi function for any integer n, so long as you know its prime factorization. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #5 20090402 01:43:13#6 20090402 01:55:21#7 20090402 09:28:02
Re: Number theoryNice #8 20090403 01:28:50
Re: Number theoryI like the Galoistheory version of Fermat’s little theorem: I’ve never seen it stated like this myself – so I claim originality for the statement of Fermat’s little theorem in this form. Last edited by JaneFairfax (20090403 01:31:06) #9 20090404 01:23:33
Re: Number theoryTreating as a polynomial over has the advantage of enabling us to prove Wilson’s theorem! Now, Fermat’s theorem in the language of Galois theory means this: Putting gives If , we get ; if , the same equation is true as . This proves Wilson’s theorem – as my friend algebraic topology points out. http://z8.invisionfree.com/DYK/index.php?showtopic=831 Last edited by JaneFairfax (20090404 11:17:38) #10 20090404 04:00:22
Re: Number theoryWilson's theorem is a nice result, and gives a good necessary and sufficient condition for prime numbers. It is however computationally inefficient for primality testing. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #11 20090411 08:28:55
Re: Number theoryWilson’s theorem appears to be something not many people try to make use of. 