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

JaneFairfax
Member
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.

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

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

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

### Re: Number theory

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

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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
Moderator
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

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

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

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

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

### 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.

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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.

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!

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 (2009-04-03 12:17:38)

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

Ricky
Moderator
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.

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

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