20160824, 14:27  #45 
Aug 2006
2^{2}×1,483 Posts 

20160824, 19:07  #46  
Dec 2008
you know...around...
599 Posts 
Quote:
Take 997#/2 ± x for instance. In the range x=[0, 2*1000], only powers of 2 are coprime to 997# (i.e. x={2,4,8,16,32,64,128,256,512,1024}, for both sides of the center number, so 20 values in total). However, outside of said range, coprimes to 997# are abundant. For every x=2*p with p>997, 997#/2 ± x is a coprime to 997#. So it is quite easy to find a merit 4gap there, but hard to find a larger gap. We have 20 coprimes to 997# in a range of 4000 numbers with x<=2000, but 270 coprimes in the range of the same size with 2000<x<=4000. Dividing 997#/2 by the next small prime 3 leaves more numbers in the range with x<=2000 coprime to 997#, all of those xvalues are 3smooth. The great benefit is that half of the numbers x=2*p outside that merit 4range get cancelled out, since before dividing by 3 they were equal to either 1 or 2 mod 3. Dividing yet again by the next small prime 5 leaves 5smooth values for x in the range x<=2000, but 1/4 of the numbers outside that range get cancelled out. So basically, it's all a tradeoff between the number of dsmooth numbers x in the expression p#/d# ± x and numbers outside the merit 4range with x>=2*p that can be covered by odd small primes q<=d, since every such small prime q takes care of about 1/(q1) of the remaining coprimes to p#. My calculations for the "effective merit" (in some other thread, also with an appropriate PARI program) depend exactly on those coprimes. 

20160825, 06:00  #47 
Dec 2008
you know...around...
599 Posts 

20160825, 09:13  #48  
"Antonio Key"
Sep 2011
UK
3^{2}·59 Posts 
Quote:
As all the gaps below p=4e18 have been enumerated we only need to check the gap for Cramér's conjecture if our measure of merit > log(4e18), that is if we ever find a merit > 42.83 Is that right? 

20160825, 16:59  #49 
"Dana Jacobsen"
Feb 2011
Bangkok, TH
2·3·151 Posts 
For the largest merits, Dr. Nicely already checks this value. If you have output in the "gap merit p1" format, this can be used to flag interesting values:
Code:
perl nE 'next unless my($g,$m)=/^(\d+)\s+(\S+)/; my $l=$g/$m; my $c=$g/$l**2; print "$c $_" if $c > 0.2;' file.txt Note the above script also works on the merits.txt file so you can examine the current values. 
20160825, 17:22  #50  
Aug 2006
2^{2}·1,483 Posts 
Quote:
Of course gathering statistics on smaller gaps is also useful! 

20160825, 19:34  #51 
"Antonio Key"
Sep 2011
UK
3^{2}·59 Posts 
As a little exercise I plotted all the results credited to myself, and all the results with merit , g/log p, greater than 30 converted to merit g/(log p)^2 the results look quite interesting.

20160825, 20:51  #52 
Jun 2003
Oxford, UK
2·13·73 Posts 
It is interesting. The first graph is more interesting I think. Something to do with size of the smallest prime in a gap and computing capability to find gaps, rather than new maths to replace CSG.
the second graph is just a function of computing power and level of effort. The 30 requirement explains the straight line. But I am not the maths bod. 
20160826, 09:23  #53  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
3^{2}×7^{2}×13 Posts 
Quote:
I understand that it makes less numbers to test past x=2000, however, I don't quite understand why as surely 1/3 of numbers are divisible by 3 anyway. I assume it is a case of k*997#/6+x has less numbers to test for x>2000 if gcd(k,3)=1. Need to think some more. 

20160827, 11:51  #54  
"Antonio Key"
Sep 2011
UK
3^{2}×59 Posts 
Quote:
Interesting discontinuity at the point where the linear search (up to 4e18 and gaps less than approximately 1346) ends and all other results continue. Last fiddled with by Antonio on 20160827 at 12:38 

20160827, 16:51  #55 
Aug 2006
2^{2}·1,483 Posts 

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