Here is this message with my editings

(sorry Phil!)

%%%%%%%

NMBRTHRY archives -- November 2001 (#9)

Date: Wed, 7 Nov 2001 09:17:30 -0500

Reply-To: Phil Carmody <fatphil@...>

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,

wrote:

> there is mistake -

> on my side,

> my "d" is larger than yours...

> Zak

>

> --- In primenumbers@yahoogroups.com, "Ken Davis" <kraden@y...>

> > This is posted on behalf of

<seidovzf@y...>

> > richyfortythree

> > cheers

> > Ken

> > > By my calculation the smallest d for p=11 is

> > > 1536160080. Have I

> > > made a

> > > mistake?

> >

> > 1536160080 is also what I get. (Same mistake maybe?)

> >

> > Cheers

> >

> > richyfourtythree

> >

> >

> > --- In primenumbers@yahoogroups.com, "mad37wriggle"

> > <fitzhughrichard@h...> wrote:

> > >

> > > By my calculation the smallest d for p=11 is 1536160080. Have I

> > made a

> > > mistake?

> > >

> > > Richard

> > >

> > >

> > > --- In primenumbers@yahoogroups.com, "Zak Seidov"

> > > wrote:

P

> > > > 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

> > > > number of

> > > > >primes that are the same numerical[equal] distance apart?