## Re: [PrimeNumbers] FW primes in arithmetic progressions

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• ... http://primes.utm.edu/glossary/page.php?sort=DicksonsConjecture says: Dickson conjectured in 1904 that given a family of linear functions with integer
Message 1 of 4 , Jan 28, 2006
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> Off the cuff, I don't see how Dickson's cinjecture solves the
> problem as posed.

http://primes.utm.edu/glossary/page.php?sort=DicksonsConjecture says:
Dickson conjectured in 1904 that given a family of linear functions
with integer coefficients a_i >= 1 and b_i:
a_1*n + b_1, a_2*n + b_2, ... a_k*n + b_k,
then there are infinitely many integers n > 0 for which these are
simultaneously prime unless they "obviously cannot be" (that is,
unless there is a prime p which divides the product of these for all n).

Let p be a given prime. Let k=p-1. Let b_i=p and a_i=i for i=1...k.
Then the conjecture says there exists n > 0 with p-1 simultaneous primes:
1n+p, 2n+p, ..., (p-1)n+p
Together with p, this gives the wanted progression of length p starting at p.

>
> 23 56211383760397 + 44546738095860·n 0..22 16 2004
> Markus Frind, Paul Jobling, Paul Underwood
>
> but that doesn't start at 23!

That's why I wrote:
> The minimal progression for p<=17 is listed in
> http://hjem.get2net.dk/jka/math/aprecords.htm#minimalstart
>
> Existence has not been proved for any larger p.

23>17.
The AP23 at the link is red, meaning it is the best known but
(probably) not minimal.

--
Jens Kruse Andersen
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