## Re: [PrimeNumbers] Primes not in AP

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• ... An AP of primes starting at p cannot have length above p, because p would divide term p+1 which is p + a*p where a is the common difference in the AP. It
Message 1 of 2 , Mar 26, 2008
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Robert wrote:
> are there primes that are not in AP, if AP>2?

An AP of primes starting at p cannot have length above p,
because p would divide term p+1 which is p + a*p
where a is the common difference in the AP.

It follows from commonly believed conjectures like
Dickson's conjecture that every prime p is the start
of infinitely many cases of p primes in AP.
The largest p with a known case is 17 where the minimal case is
17 + 341976204789992332560*n for n = 0..16, found by Phil Carmody in 2001:
http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0111&L=nmbrthry&F=&S=&P=941

I don't think it has been proved for any prime that it is in
an AP3 without finding a specific example AP3.

--
Jens Kruse Andersen
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