number congruent to -1mod 3

Jan 2016
Show that if N is congruent to -1mod3 then there exists at least one prime factor congruent to -1mod3. Hence show there are infinitely many primes of the form 6n-1.
Aug 2017
United Kingdom
As N is -1 mod 3, each of N's prime factors must be 1 or -1 mod 3. If they were all 1 mod 3, then their product (which is N) would be 1 mod 3, which is a contradiction. So at least one of them is -1 mod 3.

For the second part, consider the usual proof that there are infinitely many primes and see if you can apply the above result.
  • Like
Reactions: 2 people