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• ## Re: [PrimeNumbers] 2 AP21 with the same difference

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• ... Great! 22 primes in a fixed pattern is impressive. ... Combining these two construction principles means that if two AP22 with the same common difference
Nov 23, 2008 1 of 8
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Jarek wrote:
> Known AP20+ with difference 43#
> First terms of progressions are given
>
> AP20 9372688136871853 (old)
> AP20 11735227242889999 (old)
> AP20 202860934798777373 (new)
> AP21 195625258610971297 (new)
> AP22 93490858594661729 (new)

Great! 22 primes in a fixed pattern is impressive.
I wrote:
> Let p+n*d for n = 0..21 be any AP22 with common difference d.
> Consider (p+10.5d), 0.5d, d, 1.5d, 2d, 2.5d, 3d.
> All positive additive combinations hit one of the 22 primes in AP.
...
> Next let p+n*d and q+n*d for n=0..15 be any two non-overlapping
> AP16 with the same common difference d. Assume p<q.
> (p+q+15d)/2 +/- (q-p)/2 is respectively q+7.5d and p+7.5d
> which are in the middle of the two AP16.
> Consider (p+q+15d)/2, (q-p)/2, 0.5d, d, 1.5d, 2d, 2.5d.
> The 32 primes are hit by the additive combinations.

Combining these two construction principles means that if two
AP22 with the same common difference were known then they
would allow 8 distinct numbers such that all 128 positive additive
combinations hit one of the 44 primes.
If p+n*d and q+n*d for n=0..21 with p<q are non-overlapping
AP22 then 8 such numbers are:
(p+q+21d)/2, (q-p)/2, 0.5d, d, 1.5d, 2d, 2.5d, 3d.

As far as I know the current record is 7 distinct numbers.

--
Jens Kruse Andersen
• ... Brilliant stuff. At first I couldn t connect the dots between primes in arithmetic progression and what I was doing, but there it is. ... If so, it s not
Nov 23, 2008 1 of 8
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--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:
>
> Jarek wrote:
> > Known AP20+ with difference 43#
> > First terms of progressions are given
> >
> > AP20 9372688136871853 (old)
> > AP20 11735227242889999 (old)
> > AP20 202860934798777373 (new)
> > AP21 195625258610971297 (new)
> > AP22 93490858594661729 (new)
>
> Great! 22 primes in a fixed pattern is impressive.
> I wrote:
> > Let p+n*d for n = 0..21 be any AP22 with common difference d.
> > Consider (p+10.5d), 0.5d, d, 1.5d, 2d, 2.5d, 3d.
> > All positive additive combinations hit one of the 22 primes in AP.
> ...
> > Next let p+n*d and q+n*d for n=0..15 be any two non-overlapping
> > AP16 with the same common difference d. Assume p<q.
> > (p+q+15d)/2 +/- (q-p)/2 is respectively q+7.5d and p+7.5d
> > which are in the middle of the two AP16.
> > Consider (p+q+15d)/2, (q-p)/2, 0.5d, d, 1.5d, 2d, 2.5d.
> > The 32 primes are hit by the additive combinations.
>
> Combining these two construction principles means that if two
> AP22 with the same common difference were known then they
> would allow 8 distinct numbers such that all 128 positive additive
> combinations hit one of the 44 primes.
> If p+n*d and q+n*d for n=0..21 with p<q are non-overlapping
> AP22 then 8 such numbers are:
> (p+q+21d)/2, (q-p)/2, 0.5d, d, 1.5d, 2d, 2.5d, 3d.

Brilliant stuff. At first I couldn't connect the dots between primes in arithmetic progression
and what I was doing, but there it is.

>
> As far as I know the current record is 7 distinct numbers.
>

If so, it's not from me. If anyone else is working on this problem I'd like to know.
Yesterday I was checking numbers up to about 600, going through about 1/50 of the
possibilities. I found one with 63 out the 64 numbers prime. Close but no cigar. And only
48 distinct primes from it. But, I found many combos generating up to 59 primes, all
distinct. So I was encouraged that I would might hit all 64 prime. So I started running my
gp pari program full tilt this morning hoping to hit the jackpot. I'm now over half way
through all the possibilities, and still no jackpot. I get the feeling I'm going to have to use
numbers up to 1000 to strike it rich. But that could take, eeek, perhaps a month of
computer time. So much for instant gratification...

At least (in the last four days) I have gp pari up and running on my two year old imac
under Leopard. Much snappier than my old faithful Pentium II Windows 98 machine. If
anyone wants to know how a total non geek got gp pari to work on a mac, feel free to
contact me.

Mark
• ... Have dual G5 mac sitting idle. Post code! Phil
Nov 24, 2008 1 of 8
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--- On Mon, 11/24/08, Mark Underwood <mark.underwood@...> wrote:
> If so, it's not from me. If anyone else is working on
> this problem I'd like to know.
> Yesterday I was checking numbers up to about 600, going
> through about 1/50 of the
> possibilities. I found one with 63 out the 64 numbers
> prime. Close but no cigar. And only
> 48 distinct primes from it. But, I found many combos
> generating up to 59 primes, all
> distinct. So I was encouraged that I would might hit all 64
> prime. So I started running my
> gp pari program full tilt this morning hoping to hit the
> jackpot. I'm now over half way
> through all the possibilities, and still no jackpot. I get
> the feeling I'm going to have to use
> numbers up to 1000 to strike it rich. But that could take,
> eeek, perhaps a month of
> computer time. So much for instant gratification...

Have dual G5 mac sitting idle. Post code!

Phil
• ... That sounds great Phil. But because I am, shall we say, modest (embarrassed is more like it) about my code I ll send it privately. :) I have to rush out
Nov 24, 2008 1 of 8
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--- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...> wrote:
>
> --- On Mon, 11/24/08, Mark Underwood <mark.underwood@...> wrote:
> > If so, it's not from me. If anyone else is working on
> > this problem I'd like to know.
> > Yesterday I was checking numbers up to about 600, going
> > through about 1/50 of the
> > possibilities. I found one with 63 out the 64 numbers
> > prime. Close but no cigar. And only
> > 48 distinct primes from it. But, I found many combos
> > generating up to 59 primes, all
> > distinct. So I was encouraged that I would might hit all 64
> > prime. So I started running my
> > gp pari program full tilt this morning hoping to hit the
> > jackpot. I'm now over half way
> > through all the possibilities, and still no jackpot. I get
> > the feeling I'm going to have to use
> > numbers up to 1000 to strike it rich. But that could take,
> > eeek, perhaps a month of
> > computer time. So much for instant gratification...
>
> Have dual G5 mac sitting idle. Post code!

That sounds great Phil. But because I am, shall we say, modest (embarrassed is more like
it) about my code I'll send it privately. :) I have to rush out right now, but should have it off
to you in about five hours. Perhaps a week ago I sent my gp pari program to someone on
the list who asked privately, but it was for six numbers generating 32 distinct primes.

BTW, the program run for 7 numbers generating 64 primes has ended after less than a day
of running. No luck. No seven numbers all under 630 generate 64 primes. But I'm
confident that going up to about 1000 (specifically, all numbers under 1050) will clinch it.

Mark
• ... Well, it was lucky and it happenned only once. ... Unfortunately I am unable to specify the progression difference in advance and THEN REASONABLY EXPECT to
Nov 25, 2008 1 of 8
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--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
<jens.k.a@...> wrote:
> Great! 22 primes in a fixed pattern is impressive.

Well, it was lucky and it happenned only once.

> Combining these two construction principles means that if two
> AP22 with the same common difference were known then ........

Unfortunately I am unable to specify the progression difference in
advance and THEN REASONABLY EXPECT to find AP22 with that difference.
Despite 58 64-bit computers working on it for 3 days now, I was unable
to find another AP22 with predefined difference. So far I am trying
differences 43# and 41#. If that fails to produce 2 AP22 with the same
difference, I will try a few other near primorial differences, e.g.
37#*p with p=43,47,53,59,61.

Known AP21 with the same difference
First terms of progressions are given

Difference 43#:
AP21 76240762416222539
AP21 195625258610971297
AP21 224957853888083671
AP22 93490858594661729

Difference 41#:
AP21 86500161134359417
AP21 138060757742265263

Jarek
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