Math Is Fun Forum

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.

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

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.

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)

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..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Number theory

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Number theory

   

   

Daniel123

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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.

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)

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)

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..."

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.