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Old 2015-09-11, 11:28   #1
mart_r
 
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Default Large Gaps >500,000

Shouldn't the Border PRP's at http://www.worldofnumbers.com/ from Patrick DeGeest et al. also be included in Dr. Nicely's list?

There's, for example, a record length of 841716 (Merit: 4.72) before and after 10^77418.
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Old 2015-09-11, 15:59   #2
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Quote:
Originally Posted by mart_r View Post
Shouldn't the Border PRP's at http://www.worldofnumbers.com/ from Patrick DeGeest et al. also be included in Dr. Nicely's list?

There's, for example, a record length of 841716 (Merit: 4.72) before and after 10^77418.
I can't see any reason why these would not be on the list, except that the discoverer has not submitted the gaps to Dr. Nicely

I don't think there is any reason why smaller merit gaps can't still be posted - nothing on the site suggests this. Personally, I won't post now unless my gap is 10 merit or better. But in 2008 I did post smaller merits than that.
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Old 2015-09-11, 18:17   #3
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They certainly could be added, and Patrick De Geest is already on Nicely's author list. I would guess the merits were just too low for them to bother with.

There sure are a lot more frequent updates now than before the threads here. I'm not sure if Antonio Key is using his own software or, since he's on this forum and started submitting after my post, using my code. Dr. Nicely may not care any more -- he used to be more strict (e.g. everyone using TOeS's codes has TOS in their abbreviated name, similar with JKA).

I'm not sure what to think of the "running code based on Perl" after my name :). I wrote GMP code plus an interface to Perl, and various scripts that use it in Perl. Perhaps everyone else should have "running code based on C, libc, libm, and GMP/gwnum" added :). Of course big kudos to Torbjörn and the other GMP devs, plus Baillie, Wagstaff, Selfridge, Crandall, Pomerance, Cohen, etc. without whom I'd have nothing. Plus encouragement of other open source devs/writers like Kim Walisch, Wojciech Izykowski, David Cleaver, Mario Roy, the Pari/GP team, CRG IV, TOeS, etc. The list could go on a long time.
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Old 2015-12-30, 17:46   #4
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I'd like to enter "nextprime(primorial(20443)/2229464046810)-precprime(primorial(20443)/2229464046810)" into WolframAlpha, but I'm afraid it would take too long...


a.k.a. prove that either my new PC or OpenPFGW was wrong - at least once!
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Old 2015-12-30, 22:00   #5
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For n=primorial(20443)/2229464046810:

With Perl/ntheory I got 13522 in 23 minutes. A merit of 0.668.

Pari/GP took 80 minutes for the same result.
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Old 2015-12-30, 22:24   #6
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Quote:
Originally Posted by danaj View Post
For n=primorial(20443)/2229464046810:

With Perl/ntheory I got 13522 in 23 minutes. A merit of 0.668.

Pari/GP took 80 minutes for the same result.
Then something must be wrong with my computation... thanks for the doublecheck!
I was reluctant to submit my results because I doubted the gap was way >4M, according to my PC.
What bounding primes do you have?
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Old 2015-12-31, 01:09   #7
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Quote:
Originally Posted by mart_r View Post
What bounding primes do you have?
n = primorial(20443)/2229464046810 = 4707 ... 301362933747165290194855805813 (8796 digits)

n - 3822 = 4707 ... 301362933747165290194855801991

n + 9700 = 4707 ... 301362933747165290194855815513

Since they're probable primes (BPSW, Frobenius, Frobenius-Underwood, Frobenius-Khashin) we certainly expect them to pass a Fermat test:


$ ./pfgw64s -q'20443#/2229464046810-3822'
PFGW Version 3.7.7.64BIT.20130722.x86_Dev [GWNUM 27.11]

20443#/2229464046810-3822 is 3-PRP! (1.6745s+0.0859s)



$ ./pfgw64s -q'20443#/2229464046810+9700'
PFGW Version 3.7.7.64BIT.20130722.x86_Dev [GWNUM 27.11]

20443#/2229464046810+9700 is 3-PRP! (1.6576s+0.0814s)
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Old 2015-12-31, 03:10   #8
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With PFGW 3.7.7 (and 3.8.0) the prevprime and nextprime use a very shallow small factor test plus David Cleaver's strong Lucas GMP code. It's a good solution for "normal" size inputs, but for large ones it is quite slow. The lack of a deep sieve plus the more expensive primality test lead to a prevprime time for this input of 88 minutes vs. 6 minutes for Perl/ntheory.

Something like Dr. Nicely's cglp4 does would be *much* faster -- do a deep sieve then call pfgw for each candidate. Using Perl/ntheory's deeper sieve combined with PFGW for the candidates would be a bit over 3 minutes: almost 2x faster at this size than my GMP BPSW and ~25x faster than PFGW 3.7.7's code.

I imagine it just isn't something used very much, and just added for convenience on small inputs. So not something we'd want to use for prime gap searches (though the underlying gwnum library would help, or calling PFGW for its fast Fermat test on a single input).
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Old 2016-01-03, 02:08   #9
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Quote:
Originally Posted by mart_r View Post
Then something must be wrong with my computation... thanks for the doublecheck!
I was reluctant to submit my results because I doubted the gap was way >4M, according to my PC.
What bounding primes do you have?
I was already half asleep when I posted this and completely failed to notice danaj's misinterpretation of the WolframAlpha formula primorial(x) where WA computes the product of the first x primes rather than the product of primes up to x...

So, to be precise: is there someone that may check if there is any prime between 230077#/2229464046810-3131794 and 230077#/2229464046810+1548362 ?
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Old 2016-01-03, 04:32   #10
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Ah, Wolfram uses p_n # instead of n#. Sorry, I did know you were giving exact Mathematica code. PFGW uses n# and p(n)#.

I ran it with Perl/ntheory using pn_primorial():

n = p_20443#//2229464046810 = 5104776 .. 363999414351 (99750 digits)

A gap of 4680156 for a number this size would be about merit 20.4. Completely believable, and enough to make it the largest gap on record.

I'm running prev_prime and next_prime, but at this size it'll take quite a while.

Last fiddled with by danaj on 2016-01-03 at 04:34
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Old 2016-01-03, 12:11   #11
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Quote:
Originally Posted by danaj View Post
A gap of 4680156 for a number this size would be about merit 20.4. Completely believable,
It wasn't only this gap that was driving me bonkers. There's yet another gap in the 3M range running parallel to it. I've been abruptly running out of primes
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