## Re: Russell E. Rierson's Question About Prime Numbers

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• Russell sent me Phil s message with p=13 and p=17! Here is this message with my editings (sorry Phil!) %%%%%%% NMBRTHRY archives -- November 2001 (#9) Date:
Message 1 of 6 , Oct 1, 2003
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Russell sent me Phil's message with p=13 and p=17!
Here is this message with my editings
(sorry Phil!)

%%%%%%%
NMBRTHRY archives -- November 2001 (#9)
Date: Wed, 7 Nov 2001 09:17:30 -0500
Sender: Number Theory List <NMBRTHRY@...>
From: Phil Carmody <fatphil@...>
Subject: Prime-producing Linear Polynomials
The linear polynomial
f(X) = dX+q
can have at most q successive terms f(0)...f(q-1) prime, (and q must
be prime for f(0) to be prime, evidently). It remains an open
question, and one with few data-points, whether such maximal q-based
q-length Arithmetic Progressions exist for every q.

q=3, d=2 yield the primes 3,5,7;
q=5, d=6 yield the primes 5,11,17,23,29;
q=7, d=150 yield the primes 7,157,307,457,607,757,907.

In 1986, Löh discovered for q=11 d=1536160080, and for q=13
d=9918821194590.

[The above was a synthesis of what Paulo Ribenboim has in The New Book
of Prime Number Records]

At the start of 2001 I started tackling the q=17 problem, and I wrote
some brute force code to attack it. The code was peppered with bugs,
and despite finding a record Cunningham Chain with the broken code
http://listserv.nodak.edu/scripts/wa.exe?A2=ind0103&L=nmbrthry&P=R423
I gave up both on the project and the code.

However, I recently encountered other tasks which seemed like a good
target for the code, and easier than the Arithmetic Progression
problem, and resolved to de-mothball the code. Within a day of
thinking this, Tom Hadley posted the very result I thought I might
search for - a minimal 15-tuple ( http://www.ltkz.demon.co.uk/kt15.txt
). Rather than kill the idea, this encouraged me!

So I (think I) fixed the bugs, and started the search again. After
roughly 5 days on a single 533MHz Alpha (21164), I found the following
result, finally toppling the 15-year-old record.

The record is now
q=17 d=341976204789992332560

And the primes are
<skip>

q=19 anyone?

(The scaling factors indicate that it might be possible with a
collaborative effort, and my code parallelises)

Thanks go to Tom Hadley for giving me a massive kick up the arse last
week and to Paul Jobling who has been a useful resource on how to
apply intelligence to brute force problems since the project began.

Phil
%%%%%%%%%%%%5

--- In primenumbers@yahoogroups.com, "Zak Seidov" <seidovzf@y...>
wrote:
> Yes, Richard and Ken,
> there is mistake -
> on my side,
> my "d" is larger than yours...
> Zak
>
wrote:
> > This is posted on behalf of
> > richyfortythree
> > cheers
> > Ken
> > > By my calculation the smallest d for p=11 is
> > > 1536160080. Have I
> > > mistake?
> >
> > 1536160080 is also what I get. (Same mistake maybe?)
> >
> > Cheers
> >
> > richyfourtythree
> >
> >
> > <fitzhughrichard@h...> wrote:
> > >
> > > By my calculation the smallest d for p=11 is 1536160080. Have I
> > > mistake?
> > >
> > > Richard
> > >
> > >
> > > --- In primenumbers@yahoogroups.com, "Zak Seidov"
<seidovzf@y...>
> > > wrote:
> > > > This is copy of my post
> > > > (sorry for those reading this twice):
> > > >
> > > > For p=11,
> > > > minimal d = 4911773580 (OEIS A088430),
> > > > and AP contains maximal number, 11, primes.
> > > >
> > > > For p=13, d should be a factor of 2310.
> > > > Who first find it (and then try 17,19,...)?
> > > > Zak
> > > >
> > > >
> > > > BTW I guess that found d is indeed minimal not unique-
> > > > there is no reason of absense of other larger d's.
> > > >
> > > >
> > > > On 28 Sep 2003, Russell E. Rierson wrote
> > > > (http://www.mathforum.org/discuss/sci.math/m/133406/540774):
> > > > >Twin primes are prime numbers such as 5 and 7, 11 and 13, 17
> and
> > 19,
> > > > >etc. These twins are only one unit apart.
> > > > >
> > > > >There are strings of prime numbers that are n-units apart:
> > > > >
> > > > >3, 5, 7, [3 prime numbers, 2 units apart]
> > > > >
> > > > >5, 11, 17, 23, 29, [5, 6 units]
> > > > >
> > > > >7, 157, 307, 457, 607, 757, 907, [7, 150 units]
> > > > >
> > > > >11... ? ...? ...? ...
> > > > >
> > > > >The question becomes: For all odd prime numbers P, are there
P
> > > > number of
> > > > >primes that are the same numerical[equal] distance apart?
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