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

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carel

Posts: 161
Registered: 12/12/04
Re: Fermat's Little Theorem
Posted: Sep 13, 2006 12:13 AM
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Sorry about previous post. Correct format below


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






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