20210106, 20:47  #122 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2^{5}×5×37 Posts 
What is the logic behind searching one side before the other? Wouldn't it be better to test the denser central region first(assuming a primorial divisor) as it will probably be needed for a record?

20210107, 01:39  #123  
"Seth"
Apr 2019
367_{10} Posts 
Quote:
1) Why do I always search downwards before upwards? 2) Why do I give up after one search? 1) I don't think it matters which direction you search or if you search the denser / less dense side first. It's possible there's some better out of order search scheme that alternates testing values on each side and from different points in the sieve but let's ignore that and focus on searching one side till we find the closest prime in that direction; Note this always takes the same expected number of PRP tests minus the small < 0.5% chance that we exceed the sieve length. My mental justification was: If one side is twice as dense and we search that side first; the expected value is nearer but the sparseness on the other side makes it still likely to find a record. If we search the sparse side first then we get a larger value but the other side is denser so it's similar. Let me know if this doesn't pass a smell test and we can try to validate by running a postfacto analysis over some run with nosideskip. for 2) I wrote a bit about this at https://github.com/sethtroisi/prime...nesidedtests 

20210107, 10:40  #124  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
13440_{8} Posts 
Quote:


20210108, 05:47  #125  
"Seth"
Apr 2019
16F_{16} Posts 
Quote:
I plotted some examples and it appears that if d the divisor has few divisors (1, 2, 3) the center is clearly less dense but as d has more divisors (6,30,210) it's not clear how much this helps. I'm not sure how to measure the improvement but I'd guess this would give another 515% improvement but would take some reasonable amount of code. I've recorded it in the low priority TODOs but I'm unlikely to write this soon (happy to help anyone interested in coding it up). 

20210123, 23:54  #126 
May 2018
233 Posts 
When d=6, the coprimes are 1 and 5, so the center is not very dense. When d=30, the coprimes are 1, 7, 11, 13, 17, 19, 23, 29, so the center is dense. When d=210, the coprimes are 1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209, so the center is very dense.

20210127, 09:06  #127  
"Seth"
Apr 2019
367 Posts 
Quote:
coprimes hides a lie with d=210 for any given m half of the coprimes will be divisible by 2, 1/3 by 3 and 1/5 by 5 so the center is still not very dense. 

20210202, 13:01  #128 
Jan 2018
61 Posts 
Hi Seth and Bobby,
when you plot the remaining candidates of a primorial value divided by 6, 30, 210, 2310 etc. you will find the remaining candidates with divider 6 are few close to the primorial center value but many a little distance from the center. The larger the divider, the more spread out the candidates are. So in a nutshell: The larger the divider, the fewer there are candidates around the center. But since the remaining candidates (say with divider 30030), are stronger candidates, chance is you will mostly find smaller gaps with the occasional find of a larger gap. There has been a post somewhere/somewhen that had a very nice graphic representation of that observation. You can easily check this yourself by writing a simple script that for instance prints the remaining candidates in an interval p(100)#/D +/ 10*P(100). after deleting all candidates that divide by the factors of the divider D. So D = 30030 has factors 2, 3, 5, 7, 11, 13 > remove all candidates that can be divided by one (or more) of these factors So D = 30 had factors 2, 3, 5 > remove all candidates that can be divided by one (or more) of these factors Compare the distribution of the candidates and you will see the pattern emerge, as I described above. Hope this is clear Michiel Last fiddled with by MJansen on 20210202 at 13:08 
20210801, 13:40  #129 
Dec 2008
you know...around...
680_{10} Posts 
I'd like to ask if anybody could confirm or refute the following, or whether I'm just misunderstanding something:
I do not believe that the upper bound of (1.12) in 1908.08613 (BanksFordTao: Large prime gaps and probabilistic models) is quite correct. Setting a_{p} = 0 mod p for all primes p <= \((\frac{y}{\log y})^{1/2}\), the error bound on the righthand side of the equation should be larger than \(O(\frac{y \log_2 y}{\log^2y})\). We do not only have the primes in the interval (0, y) to account for (given indeed by \(\frac{y}{\log y}+O(\frac{y \log_2 y}{log^2y})\)), but also the semiprimes <= y with factors larger than \((\frac{y}{\log y})^{1/2}\). These semiprimes invalidate the error bound in the given form, in the sense that their number exceeds \(O(\frac{y}{\log^2y})\) and is rather \(O(\frac{y}{\log y})\), possibly \(O(\frac{y}{\log^{3/2}y})\). 
20210801, 15:12  #130 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2^{5}·5·37 Posts 
Are you aware of the blog post for this paper? You could try asking a question there(the author might have notifications for new comments).
https://terrytao.wordpress.com/2019/...listicmodels/ 
20210801, 15:39  #131 
Dec 2008
you know...around...
1250_{8} Posts 

20210802, 13:27  #132  
Jun 2003
Oxford, UK
19·103 Posts 
Quote:
Math::Prime::Util::GMP::surround_primes($x,$y,$z); See: https://github.com/danaj/MathPrime...ter/gmp_main.c 

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