Janos Pintz survey + new results on prime gaps

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• http://arxiv.org/abs/1305.6289 Very readable and nice. Also claims a result I had conjectured here before (I had outlined a proof using Zhang, but never
Message 1 of 5 , Jul 26 5:36 PM
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http://arxiv.org/abs/1305.6289

outlined a proof using Zhang, but never checked that the proof plan could
really be carried through)
is indeed true: there are infinitely many "de Polignac numbers."

One of the most impressive claims in Pintz's paper is this. Let a "near twin"
prime p be a prime p such that p,q are two primes with p<q<p+70000000.

Then: the near-twin primes contain arbitrarily long arithmetic progressions.
• ... --So those of you who keep finding long APs with prime entries only, might instead seek long APs with twin-prime entries only (a task which ought to be
Message 2 of 5 , Jul 27 3:58 AM
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> http://arxiv.org/abs/1305.6289

> One of the most impressive claims in Pintz's paper is this. Let a "near twin"
> prime p be a prime p such that p,q are two primes with p<q<p+70000000.
>
> Then: the near-twin primes contain arbitrarily long arithmetic progressions.

--So those of you who keep finding long APs with prime entries only, might
ought to be accomplishable using similar methods).
• ... Paul Jobling found 10 twin primes in AP in 2000: https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0001&L=NMBRTHRY&F=&S=&P=8002
Message 3 of 5 , Jul 27 6:40 AM
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WarrenS wrote:
> seek long APs with twin-prime entries only

Paul Jobling found 10 twin primes in AP in 2000:
https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0001&L=NMBRTHRY&F=&S=&P=8002
http://www.primepuzzles.net/puzzles/puzz_121.htm
I don't know whether his record has been broken.

--
Jerns Kruse Andersen
• ... --oh. I was about to mention the following twin-prime arithmetic progressions which however are shorter than Jobling s: start=3, gap=2, #terms=2 start=5,
Message 4 of 5 , Jul 27 7:59 AM
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--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:
>
> WarrenS wrote:
> > seek long APs with twin-prime entries only
>
> Paul Jobling found 10 twin primes in AP in 2000:
> https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0001&L=NMBRTHRY&F=&S=&P=8002
> http://www.primepuzzles.net/puzzles/puzz_121.htm
> I don't know whether his record has been broken.
>
> --
> Jerns Kruse Andersen

--oh. I was about to mention the following twin-prime arithmetic progressions
which however are shorter than Jobling's:

start=3, gap=2, #terms=2
start=5, gap=6, #terms=3
start=5, gap=12, #terms=4
start=11, gap=13222650, #terms=5
start=17, gap=6930, #terms=5
start=11, gap=569460150, #terms=6
start=311, gap=1667820, #terms=6
start=1049, gap=643896330, #terms=7
start=191339, gap=16170, #terms=7
start=451277, gap=15248310, #terms=8
start=1767419, gap=5897430, #terms=8
start=3005291, gap=1517670, #terms=8

which I found in about 30 minutes using a very stupid search.

Jobling finds 10 APs with 10 twin primes each,
and conjectures there is a maximum length of such an AP.
But Pintz's theorem (if believe him) now disproves Jobling's conjecture
at least if the definition of "twin" prime is no longer (p, p+2) but rather
(p, p+SuitableAbsoluteConstant).

I do not think it would be too difficult to break Jobling's "10" record,
if anybody were willing to put in the effort. Jobling's search took 40 days
on a 233 MHz machine. Finding one of length 11 ought to be only
about 50 times harder using his method, but I would think there might be
better search methods partly based on sieving.
• ... --and after a few hours: start=14471111, gap=156791250, #terms=9 it looks like my stupid search method actually is comparable in efficiency to Jobling s
Message 5 of 5 , Jul 27 9:30 AM
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> > Paul Jobling found 10 twin primes in AP in 2000:
> > https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0001&L=NMBRTHRY&F=&S=&P=8002
> > http://www.primepuzzles.net/puzzles/puzz_121.htm
> --oh. I was about to mention the following twin-prime arithmetic progressions
> which however are shorter than Jobling's:
>
> start=3, gap=2, #terms=2
> start=5, gap=6, #terms=3
> start=5, gap=12, #terms=4
> start=11, gap=13222650, #terms=5
> start=17, gap=6930, #terms=5
> start=11, gap=569460150, #terms=6
> start=311, gap=1667820, #terms=6
> start=1049, gap=643896330, #terms=7
> start=191339, gap=16170, #terms=7
> start=451277, gap=15248310, #terms=8
> start=1767419, gap=5897430, #terms=8
> start=3005291, gap=1517670, #terms=8
>
> which I found in about 30 minutes using a very stupid search.

--and after a few hours:
start=14471111, gap=156791250, #terms=9

it looks like my stupid search method actually is comparable in efficiency to Jobling's search method. (Rather to my surprise, since his seemed less stupid on the face of it.)
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