Fundamental theorem of Arithmetic : Number Theory #Fundamental #theorem #of #Arithmetic : #Number #Theory

 

The Fundamental Theorem of Arithmetic : 

 Statement : Every integer n > 1 can be represented as a product of prime factors in                            only one way, apart from the order of the factors.

 Proof :

     Consider the statement

      P(n) : Every integer n >  1 can be represented as a product of prime factors in only                  one way.

     Now we prove this statement by using mathematical induction on n.

      We have 2 = 2¹

                          = A product of prime factor.

      The statement is true for n=2 i.e. p(2) is true.

      Assume that p(k) is true i.e. the statement is true for k where 1 < k  < n.

      Now we prove the statement is true for all n.

      If n is a prime number then nothing to prove.

     Suppose n is a composite number

     i.e. n = p₁ p₂ … p_s where each P_i ( 1 ≤ I ≤ s ) is prime.

    Now we prove the representation of n is unique.

    But take another representation of n as n= q₁ q₂ … q_t . where each q_i ( 1 ≤ I ≤ t ) is               prime.

     ∴ we have p₁ p₂ … p_s = q₁ .q₂ . … q_t

      Now we prove t = s and each p = some q.

     Since  p₁ p₂ … p_s = q₁ q₂ … q_t , we have  p₁ / q₁ q₂ … q_t .

ð  P₁ / q₁ or p₁ / q₂ or  … or p₁ / q_t.

 Suppose p₁ / q₁.

Since p₁ / q₁ and p₁ and q₁ are primes , we have p₁ = q₁.

∴ n = p₁ p₂ … p_s = q₁ q₂ … q_t  

                = p₁ p₂ … p_s = p₁ q₂ … q_t. ( since p₁ =q₁ )

                = p₂ p₃ …p_s = q₂ q₃ … q_t.

    Now we have p₂ / q₂ q₃ … q_t

ð  P₂ / q₂ or p₂  / q₃ or … or  p₂ / q_t

Suppose p₂ / q₂

Again since p₂ and q₂ are prime we have p₂ = q₂

On continue the above process we get p₃ =q₃ , p₄ = q₄, … p_s = q_t.

i.e each p is equal to some q.

Hence the representation of n = p₁ p₂ … p_s is a product of prime factors in one way.

                Hence the proof.

    NOTE : 

                 In the factorization of an integer n, a particular prime p may occur more                         than  once.

                 If the distinct prime factors  of n are p₁ , p₂ , … , p_s and each p_i occurs as a                    factor a_i times , 

                we can write  n  = p₁^a_1. P₂^a₂ … p_s^a_s .     ( here p^a represents p^a)

                This is called the factorization of n into prime powers.

  * Historical background of Number Theory

  *  Properties of Numbers

#Fundamental #theorem #of #Arithmetic  : #Number #Theory