state the definition of prime number and prove there are inf

state the definition of prime number and prove there are infinitely many prime numbers

Solution

A prime number is a whole number which is having only two factors i.e. 1 and the number itself.

Proof for infinitely many prime numbers

Suppose there are only finite number of primes, let n.

Let, p1, p2, p3, ... pn be the n primes.

Let, p = p1*p2*p3*...*pn + 1

Clearly, p is larger than all the prime numbers, so it does not equal one of them and p is not a prime. Thus, it must be divisble by 1, itself and atleast one of the finite primes, let pm where 1<= m <= n.

By dividing p by pm we get reminder 1, this is a contradiction. Thus our assumption that there are finite number of primes must be wrong.So, there are infinetly many number of primes.