Prove that there are infinitely many primesSolutionAnswer C

Prove that there are infinitely many primes.

Solution

Answer :

Consider the iven statement \" There are infinitely many primes \"

We prove this theorem using contradiction.

Suppose there are finetely many primes say , p₁, p₂, p₃, ... p_n.

now letus construct a new number p such that p = p₁ x p₂ x p₃ x ... x p_n+1.

Clearly , p is larger than any of the primes, so it does not equal any of them.

Since,  p₁, p₂, p₃, ... p_n constitute all primes p can not be prime.Thus it must be divisible by atleast one of our finitely many primes , say p_m ( 1 m n ) . By when we divide p by p_m we get a remainder 1 which is a contraction.

So our assumption that there are finitely many primes must be false.Thus there are infinitely many primes.