Fermat’s theorem proof

Consider the sequence a, 2a,,(p-1)a.

They are all different (modulo p), because if ma=na with 1m<np-1 then 0=a(m-n), and since  pa, we get p(m-n),  which is impossible.

Now, since all these numbers are different, the set  {a, 2a, 3a,,(p-1)a}  will have the p-1 possible congruence classes (although not necessarily in the same order) and therefore


and using  gcd((p-1)!,p)=1  we get

Title Fermat’s theorem proof
Canonical name FermatsTheoremProof
Date of creation 2013-03-22 11:46:10
Last modified on 2013-03-22 11:46:10
Owner drini (3)
Last modified by drini (3)
Numerical id 11
Author drini (3)
Entry type Proof
Classification msc 11-00
Classification msc 37B55
Related topic EulerFermatTheorem
Related topic FermatsLittleTheorem
Related topic ProofOfEulerFermatTheoremUsingLagrangesTheorem
Related topic FermatsLittleTheoremProofInductive