document.write( "Question 1030738: Prove that there are infinitely many primes of the form 6n − 1.
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Algebra.Com's Answer #645621 by ikleyn(52800) $\"\"$ $\"About$

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document.write( "Suppose there are finitely many primes of the form 6n − 1\r\n" );
document.write( "and these are p1, p2, ..., . \r\n" );
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document.write( "Take M = . \r\n" );
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document.write( "If M is a prime, we have a contradiction, because, M is of the form 6n - 1 but not on our list. \r\n" );
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document.write( "If M is not a prime, then it has some prime factors , none of which\r\n" );
document.write( "are 2, 3, p1, p2, . . . , ,  so they must be of the form  6n + 1  or  6n − 1. \r\n" );
document.write( "But if all the    are of the form  6n + 1 then their product would also have this form\r\n" );
document.write( "which M does not. Therefore,  at least one of the    is a new prime of the form 6n-1. \r\n" );
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document.write( "Thus our set was not complete, and we got a contradiction with the original assumption.\r\n" );
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document.write( "So, there are in fact infinitely many primes of this form.\r\n" );
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