My Math Forum number congruent to -1mod 3

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 February 23rd, 2019, 08:07 AM #1 Senior Member   Joined: Jan 2016 From: Blackpool Posts: 103 Thanks: 2 number congruent to -1mod 3 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.
 February 23rd, 2019, 12:16 PM #2 Senior Member   Joined: Aug 2017 From: United Kingdom Posts: 311 Thanks: 109 Math Focus: Number Theory, Algebraic Geometry 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. Thanks from topsquark and Jaket1

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