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Topic: Fermat's Little Theorem
Replies: 13   Last Post: Sep 16, 2006 4:51 AM

 Messages: [ Previous | Next ]
 carel Posts: 161 Registered: 12/12/04
Re: Fermat's Little Theorem
Posted: Sep 13, 2006 12:13 AM

Sorry about previous post. Correct format below

> Fermat stated that if p prime then p divides a^p - a where a integer.
>
>
>
> For example let p =7
>
> For a = 1 we have a^p - a = 1-1 = 0 and 7 divides 0.
>
> For a = 2 we have a^p - a = 27 - 2 = 128-2=126 and 7 divides 126
>
> For a =3 we have a^p - a = 37 - 3 = 2184 and 7 divides 2184.
>
>
>
> Fermat used induction to prove this theorem. I am going to use mod
> theorem.
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>
>
> Let == suffice for congruent.
>
> Let p be prime and let a be any integer except p
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>
>
> a.1 == x_1 mod p
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> a.2 == x_2 mod p
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> a.3 == x_3 mod p
>
> .
>
> .
>
> a.j == x_j mod p
>
> .
>
> .
>
> a.(p-1)== x_(p-1) mod p
>
>
>
> Notice that all 0 < x < p and that no 2 x are the same.
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>
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> If any two x'e the same we have that
>
>
>
> a.v == x mod p and a.w == x mod p , so a(v-w) == 0 mod p , so p dives a or
> p divides v-w, but this cannot be, so no x'e are the same.
>
>
>
> It then follows that (p-1)!a^(p-1) == (p-1)! mod p , but p does not divide
> (p-1)!
>
>
>
> So a^(p-1) == 1 mod p , so p divides a^(p-1) - 1
>
>
>
> Therefore p divides a^p - a for any integer a
>
>
>
> ____________________________________________________________
>
>

Date Subject Author
9/12/06 carel
9/13/06 Gerry Myerson
9/13/06 magidin@math.berkeley.edu
9/13/06 carel
9/14/06 magidin@math.berkeley.edu
9/14/06 Gerry Myerson
9/14/06 Lee Rudolph
9/15/06 Keith Ramsay
9/15/06 magidin@math.berkeley.edu
9/13/06 carel
9/13/06 bert
9/14/06 bert
9/15/06 Pete Klimek
9/16/06 bert