20201101, 11:36  #111 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
5,857 Posts 
It strikes me that we are sieving the wrong dimension. In A*p#/d+x we currently sieve over x. Why don't we sieve over A? For each A we will count how many sieves over x it has survived. The As with the lowest score can then be looked at in the traditional way. Generally, we have a much larger range of A than the maximum gap size so the number of sieves should be reduced. This should mean deeper sieving.

20201103, 15:46  #112  
Jun 2003
Oxford, UK
2^{2}·3·7·23 Posts 
Quote:


20201104, 11:45  #113  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
5,857 Posts 
Quote:
Code:
0% 5% 10% 15% 20% 25% 30% 1.599538e10 4.355472e08 3.659155e07 4.189505e07 1.694673e06 2.834374e06 1.046412e05 35% 40% 45% 50% 55% 60% 65% 1.260456e05 1.339614e05 1.423743e05 1.597335e05 6.638588e05 1.672100e04 1.875974e04 70% 75% 80% 85% 90% 95% 100% 6.385621e04 9.265170e04 9.913896e04 1.035973e03 1.078905e03 1.138927e03 1.744451e03 Here are the results of a regression fit predicting the count remaining as the gcd of A and the divisor varies. 11 is the best by quite a bit and then 1, 7 and 77 are fairly close together. The fit is amazingly accurate. Code:
Call: lm(formula = count ~ .  1, data = sparse[sparse$k > 0, ]) Residuals: Min 1Q Median 3Q Max 137.862 17.733 0.148 17.603 143.204 Coefficients: Estimate Std. Error t value Pr(>t) gcddiv1 2.029e+03 5.882e02 34491 <2e16 *** gcddiv2 3.314e+03 5.882e02 56349 <2e16 *** gcddiv3 2.561e+03 8.318e02 30785 <2e16 *** gcddiv5 2.166e+03 1.176e01 18411 <2e16 *** gcddiv6 4.361e+03 8.318e02 52425 <2e16 *** gcddiv7 2.032e+03 1.441e01 14105 <2e16 *** gcddiv10 3.910e+03 1.176e01 33236 <2e16 *** gcddiv11 1.963e+03 1.860e01 10554 <2e16 *** gcddiv14 3.765e+03 1.441e01 26135 <2e16 *** gcddiv15 3.153e+03 1.664e01 18952 <2e16 *** gcddiv21 2.871e+03 2.038e01 14089 <2e16 *** gcddiv22 3.571e+03 1.860e01 19199 <2e16 *** gcddiv30 5.100e+03 1.664e01 30658 <2e16 *** gcddiv33 2.693e+03 2.630e01 10239 <2e16 *** gcddiv35 2.344e+03 2.882e01 8135 <2e16 *** gcddiv42 5.042e+03 2.038e01 24743 <2e16 *** gcddiv55 2.216e+03 3.720e01 5957 <2e16 *** gcddiv66 4.798e+03 2.631e01 18240 <2e16 *** gcddiv70 4.530e+03 2.882e01 15721 <2e16 *** gcddiv77 2.048e+03 4.556e01 4495 <2e16 *** gcddiv105 3.678e+03 4.075e01 9025 <2e16 *** gcddiv110 4.285e+03 3.720e01 11518 <2e16 *** gcddiv154 4.117e+03 4.556e01 9035 <2e16 *** gcddiv165 3.420e+03 5.261e01 6500 <2e16 *** gcddiv210 5.921e+03 4.075e01 14529 <2e16 *** gcddiv231 3.096e+03 6.443e01 4806 <2e16 *** gcddiv330 5.648e+03 5.260e01 10738 <2e16 *** gcddiv385 2.485e+03 9.111e01 2728 <2e16 *** gcddiv462 5.580e+03 6.443e01 8662 <2e16 *** gcddiv770 5.004e+03 9.111e01 5493 <2e16 *** gcddiv1155 4.044e+03 1.289e+00 3138 <2e16 *** gcddiv2310 6.571e+03 1.290e+00 5094 <2e16 ***  Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 26.81 on 999967 degrees of freedom Multiple Rsquared: 0.9999, Adjusted Rsquared: 0.9999 Fstatistic: 4.568e+08 on 32 and 999967 DF, pvalue: < 2.2e16 

20201104, 12:03  #114 
"Seth"
Apr 2019
414_{8} Posts 
More on this today. I'm going to releasing my code which can do this interval is a handful of seconds.
Last fiddled with by SethTro on 20201104 at 12:04 
20201207, 11:29  #115 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
5,857 Posts 
Does a divisor not make sense when using a CRT offset?
I have minimized the remaining candidates ignoring the primes in the divisor, calculated the CRT offset and then sieved multiple ks over the same range of x for k*p#/d+CRT+x and got worse results than for just k*p#/d+x. Am I doing something wrong or does it just not make sense? Is there a way of combining the two methods usefully? 
20201207, 14:39  #116  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
16E1_{16} Posts 
Quote:


20201208, 02:35  #117 
"Seth"
Apr 2019
2^{2}·67 Posts 
I published Arxiv: "Combined Sieve Algorithm for Prime Gaps" today.
There's a write up on running the code in https://www.mersenneforum.org/showthread.php?t=26286 
20201208, 17:39  #118  
Dec 2008
you know...around...
27F_{16} Posts 
Quote:
That's amazing! Excellent work! (Now I only need to wrap my head around running the program... things like building GMP and compiling libraries is something I haven't actually done much yet ) 

20201210, 11:59  #119  
Jun 2003
Oxford, UK
2^{2}·3·7·23 Posts 
Quote:
I think you are right, over the long run, deficient primorial offsets are not as efficient as defiicent primorials around the centre point, but they still provide a pretty sparse set of nonsmooth integers in a range  enough to be interesting. Over a wider range of say 40 merit, I don't think any solution provides a good result although in theory, the offset will produce a better result than the centre based solution, but the number of nonsmooth candidates to check is still such that a result is unlikely at that level. In fact to date, only one 40 merit has been found out of the infinite number of gaps of this size (!!) But certainly, the deficient primorial offsets I am generating with ATH software produces at lot of 20 merit and I have 4 records in the last week at the 3k gap area 

20201210, 13:04  #120 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
5,857 Posts 
I can generate offsets a fair bit better than ATH's code. I have been meaning to backport my code into his.
My code follows the approach of varying the offset for each prime individually. Even one pass is pretty good and better than just trying random guesses and fiddling with the last few primes like ATH's code does. I have implemented changing 2, 3 or 4 offsets at once. This can provide small improvements but is a lot slower(especially as the number of primes increases). 
20201221, 11:06  #121 
"Seth"
Apr 2019
414_{8} Posts 
I wanted to write down a little bit of analysis I've been doing recently on the computation of Probability of Record, P(record) given a partial sieve.
This uses a number of conventions from the combined sieve. K = P#/d N = m * K for many m SL = sieve length (how far after and before N to sieve) At a high level we can estimate the P(Record) by taking the double sum, Code:
for X in {x  0 < x <= SL, (x, K) = 1} for Y in {y  SL <= y < 0, (y, K) = 1} if record[X+Y] > N: P_record += ProbFirstPrime(X) * ProbFirstPrime(Y) ProbPrimeCoprime = 1 / log(N) * Product(1  1/p, p  K) ProbFirstPrime(X) = (1  ProbPrime)^(smaller coprimes) * ProbPrime This doesn't make any use of the partial sieve but it's easy to replace {X  0 < x < SL, (x, K) = 1} with the result of the partial sieve and ProbPrimeCoPrime with ProbPrimeAfterSieve = 1 / log(N) * Product(1  1/p, p <= P_limit). The next part of the analysis is to mix and match using the partial sieve results up to SL and coprime results up to some larger limit (~30*log(N)). X = partial_sieve + larger coprimes This gives us four regions of X cross Y; partial_sieve * partial_sieve, partial * coprime, coprime * partial, coprime * coprime. I call these results "direct", "extended" (partial * coprime, coprime * partial) and "extended^2" (coprime * coprime). Note that the results from "extended^2" is the same for all m so it doesn't help with differentiating P(record) per m which is useful for various reasons. From previous work it seems optimal to use SL \(= \approx 10\log(N)\) but I was worried if P(record) could be accurately computed with SL that "small". It turns out a reasonable percentage of P(record) is calculated at this point but maybe SL \(= \approx 12\log(N)\) is more "optimal". NOTE: If you inspect the graphs you see that the P(record) is overestimated by extended & extended^2 after a bit of work I've explained this to myself as primes just larger than P# expect to divide 1/prime numbers in X but that slightly over counts because of the weird way that P# aligns small primes. I don't have the right code to validate this statement but it's my operating assumption. 
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