Wilson's Theorem : Number Theory #Wilson's #Theorem #: #Number #Theory

 

 Wilson’s Theorem :

 Statement :

 If p is a prime then (p-1)! + 1 = 0 ( mod p )

Proof :

         Suppose p is a prime number..

    For p=2, we have (p-1)! + 1 = 2 and 2 ≡ 0 ( mod 2 )

   i.e. the statement is true for p=2.

  Let p > 2 and a 𝟄 Z such that 1 ≤ a ≤ p-1.

 Since p is prime , we have (a,p) = 1 and hence the linear congruence ax ≡ 1 ( mod p) has

   unique solution, say x₀.

 Let a’ 𝟄 x₀^— and  1 ≤ a’ ≤ p-1 where x₀^— is the congruence class of x₀^.

Clearly aa’ ≡ 1 ( mod p )

a’ = a

ð  a² ≡ 1 ( mod p )

ð  p / a²-1

ð  p/(a-1) or p/ (a+1)

 

for p/(a-1) and a > 0 =>  a+1 = p

ð  a = p-1

 

∴ a’ = a

ð  either a = 1 or a= p-1

similarly if a’ ≠ a then a 𝟄 { 2,3,…,p-2}.

∴ the distinct a , a’ belong to the set { 2,3,…,p-2} containing (p-3) elements.

These (p-3) elements form (p-3)/2 pairs, such that the product of each pair ≡ 1 ( mod p ).

Multiply these (p-3)/2 congruences, we get

2.3….(p-2) ≡ 1 ( mod p )

=> 1.2.3….(p-2) (p-1) ≡ 1 (p-1) ( mod p )

=> (p-1)! ≡ (p-1) ( mod p )

=> (p-1)! + 1 ≡ p ( mod p )

=> (p-1)! + 1 ≡ 0 ( mod p ).

 

           ***  Hence The Proof ***

 

Note :

   The converse of the above theorem is also true i.e. (p-1)! + 1 ≡ 0 ( mod p ) implies p is prime.

*  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

 * Fermat's Theorem

#Wilson's #Theorem #: #Number #Theory