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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
Fermat's Little Theorem
Posted: Sep 12, 2006 11:47 PM
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Fermat's Little Theorem - Alternative Solution



Fermat stated that if p prime then p divides ap - a where a integer.



For example let p =7

For a = 1 we have ap - a = 1-1 = 0 and 7 divides 0.

For a = 2 we have ap - a = 27 - 2 = 128-2=126 and 7 divides 126

For a =3 we have ap - a = 37 - 3 = 2184 and 7 divides 2184.



Fermat used induction to prove this theorem. I am going to use mod theorem.



Let == suffice for congruent.

Let p be prime and let a be any integer except p



a.1 == x1 mod p

a.2 == x2 mod p

a.3 == x3 mod p

.

.

a.j == xj mod p

.

.

a.(p-1)== xp-1 mod p



Notice that all 0 < x < p and that no 2 x are the same.



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 a(p-1)(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 ap - a for any integer a



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