Question 1030738
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Prove that there are infinitely many primes of the form 6n - 1.
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<pre>
Suppose there are finitely many primes of the form 6n &#8722; 1
and these are p1, p2, ..., {{{p[k]}}}. 

Take M = {{{2*3*p[1]*p[2]*ellipsis*p[k] &#8722; 1}}}. 

If M is a prime, we have a contradiction, because, M is of the form 6n - 1 but not on our list. 

If M is not a prime, then it has some prime factors {{{q[i]}}}, none of which
are 2, 3, p1, p2, . . . , {{{p[k]}}},  so they must be of the form  6n + 1  or  6n &#8722; 1. 
But if all the  {{{q[i]}}}  are of the form  6n + 1 then their product would also have this form
which M does not. Therefore,  at least one of the  {{{q[i]}}}  is a new prime of the form 6n-1. 

Thus our set was not complete, and we got a contradiction with the original assumption.

So, there are in fact infinitely many primes of this form.
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