Prove that there are infinitely many primes of the form 6k + 5. That is, consider the primes which has a remainder 5 when divided by 6. Prove that there are infinitely many such primes.

Solution

Note that if we multiply 2 numbers of the form 6k + 1 together, we get another number of the same form

If there were only finitely many primes of the form 6k + 5, say

p0 = 5 < p₁ < p₂ < · · · < p_n.

Consider the integer

N = 6p₁p₂ · · · p_n + 5.

Clearly N > 1 is not divisible by 2 and the note with which we began this solution implies that N has at least one prime divisor p of the form 6k + 5. If p = 5, we get that

5|N − 5 = 6p₁p₂ · · · p_n,

while, if p > 5, we have that

5|N − 6p₁p₂ · · · p_n = 5.

In either case, we have a contradiction.