There Are Infinite Numbers Of Primes

Proof:

let us assume that there are finite numbers of primes,                                                         
       say it be  P₁,P₂,P₃,_{.............}P_K

       let  N=  P₁P₂P_{3.............}P_K + 1
       If N is a prime number and  N > P_i,
                    where i=1,2,3,4.............k
then it contradicts our assumption that we have finite number of primes.

So if N is a composite number, 
let P_j will divide N for some 1<j<k
=>P_j|N
=>N=P_j*t , where "t" is an integer
=>P₁,P₂,P₃,_{.............}P_K =P_j*t
=>P₁P₂P_3.....P_j......P_K + 1 = P_j*t

Devide both sides by P_j
=>(P₁P₂P_3.....P_j.......P_K + 1)/P_j  = t

=>P₁P₂P_3.....Pj-1.P_j+1.......P_K + 1/P_{j =   t}

since "t" is an integer so
P₁P₂P_3.....P_j-1.P_j+1.......P_K + /P_j will be an integer. 
which is not possible
So N must be a prime number
this is a contradiction to our assumption that N is composite.

=> N = P₁P₂P_{3.............}P_K + 1 is also a prime.
Hence there are infinite number of primes.


_____________________________________
Watch Video

____________________________________________________   
For GK Updates visit: The Infinite Studies