Thread: Prove that there are infinitely many primes in the form 6k+5

Prove that there are infinitely many primes in the form 6k+5 where k is a positive integer.

thanks!

Originally Posted by VinceW

Prove that there are infinitely many primes in the form 6k+5 where k is a positive integer.

thanks!

Notice that:

(6t+1)(6k+1)=6(6tk+t+k)+1

Now, suppose that there is finite numbers of prime of the form: 6k+1 and they are: p_1,p_2,...,p_n.

N=6p_1p_2...p_n-1=5(mod 6)

All the prime factor of N are different from p_1,p_2,...,p_n and they are not 2 or 3, N is also is of the from 6k+1 - it's a contradiction!