Prime Gap Theory

[CRGreathouse]

Originally Posted by Antonio

What is the (Log p)^2 merit useful for?

The primary purpose would be showing that the strong form of Cramér's conjecture is false.

[mart_r]

Originally Posted by henryzz

One question that has been bugging me for a bit is why do we usually divide by a small primorial? When you get to larger numbers the size of the large primorial isn't an issue. Does it affect the distribution of gaps? One thing I noticed with large primorials is that the largest gaps after sieving are a little away from the central point.
What is the best way to chose which primorial to divide by?

It all comes from the coprimes to a large primorial.
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 4-gap 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 x-values are 3-smooth. The great benefit is that half of the numbers x=2*p outside that merit 4-range 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 5-smooth 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 trade-off between the number of d-smooth numbers x in the expression p#/d# ± x and numbers outside the merit 4-range 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/(q-1) 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.

[mart_r]

Originally Posted by mart_r

My calculations for the "effective merit" (in some other thread, also with an appropriate PARI program) depend exactly on those coprimes.

I've just noticed, it's on the very first page of this same thread.

[Antonio]

Originally Posted by CRGreathouse

The primary purpose would be showing that the strong form of Cramér's conjecture is false.

Ok so, if I've understood this correctly, we need to raise a flag if the merit we normally use ever exceeds log(p) [i.e. g/log(p) > log(p)].

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?

[danaj]

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:

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

Even with the low threshold it's unlikely to show much (mine are all for gaps <= length 4404). Values above 0.5 are probably interesting. The only one above 2000 is Spielauer's gap of 2258 (merit 33.87) with a value of 0.508. Above 0.7 seems worthy of comment, but at that point the merit will be high enough to already cause attention.

Note the above script also works on the merits.txt file so you can examine the current values.

[CRGreathouse]

Originally Posted by Antonio

Ok so, if I've understood this correctly, we need to raise a flag if the merit we normally use ever exceeds log(p) [i.e. g/log(p) > log(p)].

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?

Yes, precisely.

Of course gathering statistics on smaller gaps is also useful!

[Antonio]

Originally Posted by CRGreathouse

Yes, precisely.

Of course gathering statistics on smaller gaps is also useful!

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.

[robert44444uk]

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.

[henryzz]

Originally Posted by mart_r

It all comes from the coprimes to a large primorial.
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 4-gap 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 x-values are 3-smooth. The great benefit is that half of the numbers x=2*p outside that merit 4-range 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 5-smooth 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 trade-off between the number of d-smooth numbers x in the expression p#/d# ± x and numbers outside the merit 4-range 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/(q-1) 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.

So increasing d makes reaching 4 merit harder but makes extending beyond that easier.
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.

[Antonio]

Originally Posted by Antonio

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.

Here is a plot for the Cramér-Shanks-Granville ratio for all gaps in Nicely's current merit.txt file.

Interesting discontinuity at the point where the linear search (up to 4e18 and gaps less than approximately 1346) ends and all other results continue.

[CRGreathouse]

Originally Posted by Antonio

Interesting discontinuity at the point where the linear search (up to 4e18 and gaps less than approximately 1346) ends and all other results continue.

Great plot! That suggests that we have a lot more record gaps to find, the best ones are still hidden.