Posted by **Andy** on Thursday, December 23, 2010 at 2:36am.

A student claims that every prime greater than 3 is a term in the arithmetic sequence whose nth term is 6n + 1 or in the arithmetic sequence whose nth term is 6n - 1. Is this true? if so, why?

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**drwls**, Thursday, December 23, 2010 at 3:52am
Yes, it is true.

All PRIME numbers above 3 are of the form 6n − 1 or 6n + 1, because all other numbers are divisible by 2 or 3.

That is, the other numbers above 3 (that cannot be written as 6n-1 or 6n+1) can be written 6n, 6n+2, 6n+3, or 6n+4. All of those are divisible by 2 or 3 and therefore cannot be prime.

(6n+5 can be written 6(n+1)-1 where n+1 is the next integer)

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