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#1 2009-03-30 10:36:21

JaneFairfax
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Number theory





This is something Iíve just read about in H.E. Roseís A Course in Number Theory. The proof is remarkably simple. smile

Last edited by JaneFairfax (2009-03-30 10:36:46)


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#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. smile


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#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)


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#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

JaneFairfax
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Re: Number theory











smile


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A: Click here for answer.
 

#6 2009-04-02 01:55:21

JaneFairfax
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Re: Number theory



   





   


smile


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#7 2009-04-02 09:28:02

Daniel123
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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.

 

#8 2009-04-03 01:28:50

JaneFairfax
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Re: Number theory

I like the Galois-theory version of Fermatís little theorem:




Iíve never seen it stated like this myself Ė so I claim originality for the statement of Fermatís little theorem in this form. tongue

Last edited by JaneFairfax (2009-04-03 01:31:06)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#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 Wilsonís theorem! big_smile

Now, Fermatís theorem in the language of Galois theory means this:

   


Putting
gives

   


If
, we get
; if
, the same equation is true as
.
This proves Wilsonís theorem Ė as my friend algebraic topology points out. tongue

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

Last edited by JaneFairfax (2009-04-04 11:17:38)


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A: Click here for answer.
 

#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

Wilsonís 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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A: Click here for answer.
 

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