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 2021-12-14, 17:17 #1 robert44444uk     Jun 2003 Oxford, UK 2,039 Posts Median gaps We know that the average gap g following a prime p is given as g/ln(p). Hence a g/ln(p) =1 is a merit of 1. However, over a range of primes there is a predominance of smaller gaps and a long tail of larger gaps, as well as a slight predominance of gaps of length 3#= 6; 5# = 30. Instead of average, I have been looking at the predominance of the smaller gaps (merit<1) over the larger ones and also at the median gap of a range of gaps, to find out what merit that has. I found some surprising (at least to me) results. Take a range of, say 10001 consecutive primes, and the 10000 gaps that ensue. Also find out how many of the 5000 gaps are greater than 1 merit. Then, starting with the last prime from the first range, look at the next 10000 primes and repeat etc. The attached graphs show the result for the first 15000 ranges, so up to p= 3121456873. The total gaps > merit = 1 is the y axis, and the consecutive group of 10000 gaps are shown on the x axis. x=1 is the first group of 10000 gaps. There are distinct bands of results, which surprised me. The fractures are at the points when p is nearest to e^(2n), where n is an integer. The second pair of graphs show more detail about the lower levels of n. Here I took ranges of 500 gaps, up to p= 23887993. The first graph B is a corollary of graph A, but smaller ranges of gaps. Graph C is harder to interpret. It shows the merit of the median in the range of ordered gap merits within a range of merits - in this case a range of 500 merits, so the data is the same as for Graph B. At ay stage there appear to be discrete possibilities for the values of the median prime fitting either one of two curves, for example curve B or C, and another range shows possible curve C or D. The curves appear to peter out. The ordered merits must be choppy, with a potential break around the median. I will have to look at some of the detailed data for a given ordered range of gaps. Attached Thumbnails
 2021-12-14, 17:36 #2 rudy235     Jun 2015 Vallejo, CA/. 21118 Posts Nice work Robert!
2021-12-15, 09:16   #3
robert44444uk

Jun 2003
Oxford, UK

2,039 Posts

Quote:
 Originally Posted by robert44444uk Instead of average, I have been looking at the predominance of the smaller gaps (merit<1) over the larger ones and also at the median gap of a range of gaps, to find out what merit that has. I found some surprising (at least to me) results. Take a range of, say 10001 consecutive primes, and the 10000 gaps that ensue. Also find out how many of the 5000 gaps are greater than 1 merit. Then, starting with the last prime from the first range, look at the next 10000 primes and repeat etc. The attached graphs show the result for the first 15000 ranges, so up to p= 3121456873. The total gaps > merit = 1 is the y axis, and the consecutive group of 10000 gaps are shown on the x axis. x=1 is the first group of 10000 gaps. There are distinct bands of results, which surprised me. The fractures are at the points when p is nearest to e^(2n), where n is an integer.
Actually it is easy to understand what is happening in Graph A. Take the fracture that occurs at p = e^20. A merit of 1 at prime p means that the average gap g is merit*ln(e^20) = 20. Hence above p=e^20 a gap of 20 no longer has a merit >1 and hence the results on a range will be depressed by the exclusion of all of the gaps = 20 within it, whereas just one prime less than e^20 will include gaps = 20 in its results.

Last fiddled with by robert44444uk on 2021-12-15 at 09:21

2021-12-15, 11:00   #4
robert44444uk

Jun 2003
Oxford, UK

2,039 Posts

Quote:
 Originally Posted by robert44444uk Graph C is harder to interpret. It shows the merit of the median in the range of ordered gap merits within a range of merits - in this case a range of 500 merits, so the data is the same as for Graph B. At ay stage there appear to be discrete possibilities for the values of the median prime fitting either one of two curves, for example curve B or C, and another range shows possible curve C or D. The curves appear to peter out. The ordered merits must be choppy, with a potential break around the median. I will have to look at some of the detailed data for a given ordered range of gaps.
Just looking at the middle curve C values in the graph C in the first post...when plotted against x = ln(p), rather than the x = series number gives a pretty decent match for the formula y = 11.955*ln(p)^-1
Attached Thumbnails

 2021-12-21, 17:32 #5 robert44444uk     Jun 2003 Oxford, UK 37678 Posts As well as looking at the median merits for arrays of primes from x=3 to x=1e7, where x represents the larger prime in the array 2..x, I have also calculated the median gap. The first few arrays show variable results but it then settles into long ranges of arrays of primes with the same median gap. Up to x=53, the results are as follows: Please note that if the array has an odd number of primes and hence an even number of gaps, then the median is the mean of the two gaps in the centre of the array. Hence odd "gaps" can be the median, and even 1.5 ! Code: 3 1 5 1.5 7 2 11 2 13 2 17 2 19 2 23 2 29 4 31 2 37 4 41 3 43 4 47 3 53 4 Then 4 dominates up to x=563, before 6 kicks in for x=569, 4 again at x=571, 5 at 587 From 593 to 38821 6 is the median. It then oscillates for a while between 7 and 8, with 7s at 38833, 38851, 38867, 38923, 39119, 39139, 39161, 39181 before a run of 8s from 39191 to 239139. It then oscillated between 8 and 10 until 239557 after which it 10 until 3798071. More oscillations between 10 and 12, before a long run of 12 from 3799469. I have yet to determine the swap over to 14 as the median gap. Last fiddled with by robert44444uk on 2021-12-21 at 17:32
 2021-12-22, 09:37 #6 robert44444uk     Jun 2003 Oxford, UK 2,039 Posts I have calculated the median gap merits in ordered arrays of gaps between primes 2 to p(x), where x represent primes up to 1e7. The results are shown in the attached graph. Also to note, earlier graphs can be found here: https://www.mersenneforum.org/showpo...5&postcount=13 Attached Thumbnails

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