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## Re: Some drag-racing progress...

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• ... IMHO, the sprint start will provide the most efficient route to providing the top candidates. I am not certain what the maximum # of primes by n=10,000
Message 1 of 10 , Oct 29, 2011
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>
> --- In primenumbers@yahoogroups.com,
> "robert44444uk" <robert_smith44@> wrote:
>
> > The last column refers to the Payam muliplier
>
> For Payamic progress on k*10^n-1, see
> with the record-holder at
>
> Note that this is not drag racing, in its strict sense;
> Makoto Kamada erects no artificial finishing tape.
> It makes sense to have a sprint start, as in a drag race.
> But it's also a good idea to keep k < 2^63, so that you
> can use NewPgen and PFGW efficiently when the ultra-marathon
> stage takes over, with n > 200000, fit for Prime Page entry.
>
> Best wishes
>
> David
>

IMHO, the "sprint start" will provide the most efficient route to providing the top candidates.

I am not certain what the maximum # of primes by n=10,000 is for any k such that k<2^63, but it is possibly about 95 for base 2. This is already 26 primes behind the current leader, and 15 behind the sorts of k that I can find weekly running my "sprint start" software, and 10 behind those candidates which I find every day.

Regards

Robert
• ... Two years ago I did Sierpinski and Riesel drag racing , for both normal and dual forms. I investigated ALL k
Message 2 of 10 , Oct 30, 2011
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--- In primenumbers@yahoogroups.com, "robert44444uk" <robert_smith44@...> wrote:
>
> IMHO, the "sprint start" will provide the most efficient route to providing the top candidates.
>
> I am not certain what the maximum # of primes by n=10,000 is for any k such that k<2^63, but it is possibly about 95 for base 2. This is already 26 primes behind the current leader, and 15 behind the sorts of k that I can find weekly running my "sprint start" software, and 10 behind those candidates which I find every day.
>
> Regards
>
> Robert
>

Two years ago I did Sierpinski and Riesel "drag racing", for both normal and dual forms.
I investigated ALL k < 10^9, and used the sprint-start performance (up to n=100) to select the best candidates, then to 1000 for the next cut, and so on.
Here are the winning counts of primes, for three ranges of n.

Sierpinski k*2^n+1:-
Best to n=100
k=232241655=3*5*19*814883 31
Best to n=1000
k=101822175=3^2*5^2*7*13*4973 54
k=902210505=3*5*7*8592481 54
Best to n=10000
k=78697515=3*5*13*403577 79
k=902210505=3*5*7*8592481 79

Riesel k*2^n-1:-
Best to n=100
k=201434145=3*5*11*29*43*89 30
Best to n=1000
k=144061665=3*5*11*137*6373 54
Best to n=10000
k=80932995=3^2*5*11*13*12577 83

Dual Sierpinski 2^n+k:-
Best to n=100
k=454245=3*5*11*2753 37
k=130988565=3^2*5*19*23*6661 37
Best to n=1000
k=321078615=3*5*11*13*181*827 67
Best to n=10000
k=321078615=3*5*11*13*181*827 107
and continuing this "champion of champions" up to n=100000 gives the exceptionally high count of 134.

Dual Riesel 2^n-k:-
Best to n=100
k=9548565=3*5*13*23*2129 38
k=688366965=3*5*11*13*269*1193 38
Best to n=1000
k=485186295=3*5*11*2940523 69
Best to n=10000
k=527175=3^3*5^2*11*71 96

It is interesting that the ONLY decently performing k without 3 as a factor was the following, with Sierpinski form k*2^n+1:-
k=922035=5*11*17*1061
To n=100 19
To n=1000 50
To n=10000 69

It seems that the dual forms tend to have rather higher prime densities.
I assume this can be accounted for by them being about a factor k smaller.
Anyone like to quantify this, using PNT?

Mike
• ... Hi Mike, I would concur duals are slightly more dense. I did not do much on this back in the day, but the following were the best I found on the Sierpinski
Message 3 of 10 , Oct 30, 2011
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--- In primenumbers@yahoogroups.com, "mikeoakes2" <mikeoakes2@...> wrote:

>
> Dual Sierpinski 2^n+k:-
> Best to n=100
> k=454245=3*5*11*2753 37
> k=130988565=3^2*5*19*23*6661 37
> Best to n=1000
> k=321078615=3*5*11*13*181*827 67
> Best to n=10000
> k=321078615=3*5*11*13*181*827 107
> and continuing this "champion of champions" up to n=100000 gives the exceptionally high count of 134.
>
> Dual Riesel 2^n-k:-
> Best to n=100
> k=9548565=3*5*13*23*2129 38
> k=688366965=3*5*11*13*269*1193 38
> Best to n=1000
> k=485186295=3*5*11*2940523 69
> Best to n=10000
> k=527175=3^3*5^2*11*71 96

>
> It seems that the dual forms tend to have rather higher prime densities.
> I assume this can be accounted for by them being about a factor k smaller.
> Anyone like to quantify this, using PNT?
>
> Mike
>
Hi Mike, I would concur duals are slightly more dense. I did not do much on this back in the day, but the following were the best I found on the Sierpinski side

Using M terminology where M(x-1) = multiple of primes up to x from the list 3,5,11,13,19,29,37,53,59,61,67,83,101,107

k=69338767*M(28) 99 primes by n=5650 or 99/5650
k=2852139845*M(36) 112/6000
k=132233299377*M(52)147/40917
k=19379084737*M(58) 127/20000
k=2231109458141*M(67) 158/59542
k=835387103*M(82)56/500
k=157873677392081*M(106)135/32778

Best to 100 primes: possibly a shade over n=3000, pretty similar to the Proths: k=21385711867*M(37) had 99/3000, don't have details after that.

The best performer above, the M(67) candidate compares very favourably to the best at 158 primes for proths of n=164463 for Riesel side and n=95487 for Sierpinski side.

Best to 1000: 2852139845*M(53) 82/1000, far above Proth best of 73

Best to 100: 32 by several candidates, behind the Proth best of 34

All th dual records can be improved significantly. I have checked about 1 trillion (!) k values _ much more work needed for the duals

regards

robert
• ... Ugh, keyboard issues. I checked about 1 trillion (!) k values of payam proths, mainly on the Riesel side, but only a few million for the duals.
Message 4 of 10 , Oct 30, 2011
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> All th dual records can be improved significantly. I have checked about 1 trillion (!) k values _ much more work needed for the duals
>
> regards
>
> robert
>

Ugh, keyboard issues. I checked about 1 trillion (!) k values of payam proths, mainly on the Riesel side, but only a few million for the duals.
• ... Phil A further advance, on the Riesel side, just failing at 75 primes: 74 972 147707435198851 R 52
Message 5 of 10 , Dec 14, 2011
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--- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...> wrote:
>
> --- On Thu, 10/27/11, robert44444uk <robert_smith44@...> wrote:
> > Phil Carmody <thefatphil@> wrote:
> > > Firstly, Jack, can you update Carlos Rivera with your target values,
> > > otherwise we don't know exactly what we're trying to beat!
> > >
> > > In response to Jack's "Any of these milestones are exceptional", with
> > > reference to 56 primes below 1000, I now have a number with 64 primes
> > > before n=1000. Is that some kind of record? (Weight=4.786)
> > > (no 63s, 2 62s, 4 61s, 7 60s, 14 59s, 22 58s, 29 57s, not bad for 24 hours
> > > work on a 5 year old machine.)
> > >
> > > Sure, it's  a 25-digit k, which detracts from the achievement (or
> > > does it?), but even with numbers of that size I believe that >60
> > > primes, in particular 64, before n=1000 must still be somewhat of
> > > a rarity.
> > >
> > > I just don't have the 'feel' of how to measure these things yet.
> >
> > For the record, the best performing Riesel I have found at n=1000 is
> > k=19122572047641*3*5*11*13*19*29*37*53 with 73 primes, the 73rd is at n=963
>
> Stunning!
>
> I don't know how up-to-date my database is, as I didn't fully get the website up and running whilst I was still priming, but the best I have to hand are the following:
>
> mysql> select * from records left join candidate on records.cand=candidate.id where n<=1000 && p>=70;
> +------+----+-----+------+--------+--------------+---------------+------+------+
> | cand | p | n | id | finder | k | m | plus | dual |
> +------+----+-----+------+--------+--------------+---------------+------+------+
> | 16 | 70 | 847 | 16 | g106 | 792030929331 | 2317696095 | 1 | 0 |
> | 205 | 70 | 956 | 205 | pcrc | 55120464273 | 8341388245905 | -1 | 0 |
> +------+----+-----+------+--------+--------------+---------------+------+------+
>
> Which is odd, as the m for the +1 case is the same small prime multiplier as yours, so my database has its signs all wrong (fortunately an easy fix).
>
> I did think that I broke clear of 70, but at the moment I have no proof of that. (And if I did, it was probably only 1 or 2 more.) I don't even know where all my files for that search even went. They were large enough, I may have just binned them :-(
>
> Good work, Robert!
>
> Phil
>

Phil

A further advance, on the Riesel side, just failing at 75 primes:

74 972 147707435198851 R 52
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