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February 23rd, 2019, 08:07 AM  #1 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 104 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 6n1.

February 23rd, 2019, 12:16 PM  #2 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 313 Thanks: 112 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. 

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1mod, congruent, number 
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