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#1 2009-03-29 11:36:21

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

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. smile

Last edited by JaneFairfax (2009-03-29 11:36:46)


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#2 2009-03-29 13:01:35

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Number theory


The proof relies on the multiplicative properties of the sigma function above. smile


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#3 2009-03-31 06:02:39

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Number theory


We have this interesting little result:

[align=center]

[/align]

The proof is only a few lines long. Also:

Last edited by JaneFairfax (2009-03-31 10:37:26)


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#4 2009-03-31 10:31:32

Ricky
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Registered: 2005-12-04
Posts: 3,791

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..."

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#5 2009-04-01 02:43:13

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Number theory



smile


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#6 2009-04-01 02:55:21

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Number theory

   


   

smile


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#7 2009-04-01 10:28:02

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Re: Number theory

Nice smile

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.

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#8 2009-04-02 02:28:50

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

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. tongue

Last edited by JaneFairfax (2009-04-02 02:31:06)


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#9 2009-04-03 02:23:33

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Number theory


Treating
as a polynomial over
has the advantage of enabling us to prove Wilson’s theorem! big_smile

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. tongue

http://z8.invisionfree.com/DYK/index.php?showtopic=831

Last edited by JaneFairfax (2009-04-03 12:17:38)


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#10 2009-04-03 05:00:22

Ricky
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Registered: 2005-12-04
Posts: 3,791

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..."

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#11 2009-04-10 10:28:55

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Number theory

Wilson’s theorem appears to be something not many people try to make use of. sad

For example, http://www.mathhelpforum.com/math-help/ … ility.html.


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