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#1 2009-03-30 10:36:21
- JaneFairfax
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Number theory
This 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 (2009-03-30 10:36:46)
#2 2009-03-30 12:01:35
- JaneFairfax
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Re: Number theory
The proof relies on the multiplicative properties of the sigma function above.
#3 2009-04-01 05:02:39
- JaneFairfax
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Re: Number theory
We have this interesting little result:
The proof is only a few lines long. Also:
Last edited by JaneFairfax (2009-04-01 09:37:26)
#4 2009-04-01 09:31:32
- Ricky
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Re: Number theory
That
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 2009-04-02 01:43:13
#6 2009-04-02 01:55:21
#7 2009-04-02 09:28:02
- Daniel123
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Re: Number theory
Nice
I like the combinatorial proof of Fermat's Little Theorem, which considers the number of bracelets that can be made from 'p' beads of 'a' different colours.
#8 2009-04-03 01:28:50
- JaneFairfax
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Re: Number theory
I like the Galois-theory 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 (2009-04-03 01:31:06)
#9 2009-04-04 01:23:33
- JaneFairfax
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Re: Number theory
Treating 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 (2009-04-04 11:17:38)
#10 2009-04-04 04:00:22
- Ricky
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Re: Number theory
Wilson'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 2009-04-11 08:28:55
- JaneFairfax
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Re: Number theory
Wilson’s theorem appears to be something not many people try to make use of. 
For example, http://www.mathhelpforum.com/math-help/ … ility.html.