Fermat's Theorem : Number Theory #Fermat's #Theorem : #Number #Theory

Fermat's Theorem :

Statement : 

                     If  p is a prime and (a,p) = 1 then a^p-1 ≡ 1 (mod p).

Proof :

               Suppose p is a prime number and (a,p) = 1.

        Since (a,p) = 1 then the numbers a,2a,3a,…,(p-1) are divided by p and the remainders are 1,2,3 …,                 (p-1) ; not necessarily in this order.

        Let a ≡ r₁ ( mod p), 2a ≡ r₂ ( mod p) , … , (p-1)a ≡ r_p-1 ( mod p)

        Since r₁ , r₂ , … r_p-1 are remainders obtained when a,2a,…,(p-1)a are divided by p.

        ∴ r_1. r₂ . … .r_p-1 = 1 . 2 .3 …. . (p-1).

       Multiplying the above congruent relations : a . 2a. … . (p-1) ≡ r₁r₂…r_p-1 ( mod p )

ð  {1.2….(p-1) ≡ 1.2….(p-1) (mod p)

ð       (p-1)! a^p-1 ≡ (p-1)!(mod p)

ð                  a^p-1 ≡ 1 ( mod p)    | since p is prime and                                                                     (p,1)=(p,2)=… = (p,p-1) = 1.

** Hence The Proof  **

*  Dirichlet Multiplication

 * Euclidean Algorithm

 * Fundamental Theorem of Arithmetic

 *  Properties of Numbers

 * Historical Introduction to Number Theory 

 * GCD of morethan 2 numbers

 *   The Mobius Function 𝝻 ( n ) .

 *  The Euler Totient Function 

 * Formal Power Series 

 * Liouville’s function λ(n) 

  * Congruences

* Wilson's Theorem 

                                                

#Fermat's #Theorem : #Number #Theory