mersenneforum.org Long ranges of gaps larger than a given value
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2021-12-20, 21:52   #12
mart_r

Dec 2008
you know...around...

22×5×37 Posts

Quote:
 Originally Posted by robert44444uk Checked up to p=1001200549 and a second run of 15 found but no run of 16 yet. Because the median merit following a prime is only about 0.7 merit - i.e. 0.7*g/ln(p) - there may be a case for saying that arbitrarily long runs of gaps with merit >1 do not occur. Interest to hear from others.

Well you already found 46318910903 (length 16), 103947089669 (17 and 18), and 184722051989 (19 and 20, the latter almost being of length 23), from your data in the OP at x=30. And those are probably not minimal, the columns for x=22..28 are missing - with log(184722051889) = 25.942..., x=26 would suffice here.

I do expect arbitrarily long runs of gaps with merit larger than a given value. It has long been proven that the merit of a gap between two primes can become arbitrarily large, so I would expect the same to be true for a series of consecutive gaps with at least a given merit each.

 2021-12-21, 09:39 #13 robert44444uk     Jun 2003 Oxford, UK 2,039 Posts Here is an interesting pair of graphs - they shows he median gap of a range of all gaps for primes 2 to x, with x up to 999983. The second graph ignores the results up to x=4409 so as to see the fine detail of the later ranges. I plan to carry the range up to prime range from 2 to 1e7, although the work is relatively intensive on resources it does look as if the median gap is on the rise again at 1e6 and it is worth looking at this. Attached Thumbnails     Last fiddled with by robert44444uk on 2021-12-21 at 09:42
2021-12-21, 09:40   #14
robert44444uk

Jun 2003
Oxford, UK

2,039 Posts

Quote:
 Originally Posted by mart_r Well you already found 46318910903 (length 16), 103947089669 (17 and 18), and 184722051989 (19 and 20, the latter almost being of length 23), from your data in the OP at x=30. And those are probably not minimal, the columns for x=22..28 are missing - with log(184722051889) = 25.942..., x=26 would suffice here. I do expect arbitrarily long runs of gaps with merit larger than a given value. It has long been proven that the merit of a gap between two primes can become arbitrarily large, so I would expect the same to be true for a series of consecutive gaps with at least a given merit each.
That does rather look like the case - I retract my earlier statement.

2021-12-21, 14:46   #15
robert44444uk

Jun 2003
Oxford, UK

2,039 Posts

Quote:
 Originally Posted by robert44444uk Here is an interesting pair of graphs - they shows he median gap of a range of all gaps for primes 2 to x, with x up to 999983. The second graph ignores the results up to x=4409 so as to see the fine detail of the later ranges. I plan to carry the range up to prime range from 2 to 1e7, although the work is relatively intensive on resources it does look as if the median gap is on the rise again at 1e6 and it is worth looking at this.
Please use the thread "Median Gaps" for further postings on this theme.

2021-12-23, 18:38   #16
Bobby Jacobs

May 2018

25810 Posts

Quote:
 Originally Posted by robert44444uk Min merit >5 Code: Min merit: 5 Run of merits: 2 Initial prime: 10938023 Following gaps: 102 96 Merits: 6.293287056 5.923090292 Min merit: 5 Run of merits: 3 Initial prime: 1440754039 Following gaps: 122 130 108 Merits: 5.785162115 6.164516983 5.12129101
A gap of 102 after 10938023 would make the next prime 10938125. That is obviously not prime.

2021-12-23, 22:43   #17
Dr Sardonicus

Feb 2017
Nowhere

5·1,153 Posts

Quote:
Originally Posted by Bobby Jacobs
Quote:
 Originally Posted by robert44444uk Min merit >5 Code: Min merit: 5 Run of merits: 2 Initial prime: 10938023 Following gaps: 102 96 Merits: 6.293287056 5.923090292 Min merit: 5 Run of merits: 3 Initial prime: 1440754039 Following gaps: 122 130 108 Merits: 5.785162115 6.164516983 5.12129101
A gap of 102 after 10938023 would make the next prime 10938125. That is obviously not prime.
Gotta be a minor oopsadaisy. Let's see here... Ahh, here we go!
Code:
? precprime(10938023-1)
%1 = 10937921
? nextprime(10938023+1)
%2 = 10938119
?
So, 10938023 is 102 more than the preceding prime, and 96 less than the next prime.

2021-12-24, 08:40   #18
robert44444uk

Jun 2003
Oxford, UK

2,039 Posts

Quote:
 Originally Posted by Dr Sardonicus Gotta be a minor oopsadaisy. Let's see here... Ahh, here we go! Code: ? precprime(10938023-1) %1 = 10937921 ? nextprime(10938023+1) %2 = 10938119 ? So, 10938023 is 102 more than the preceding prime, and 96 less than the next prime.
I'm no programmer, Sorry for this. What is wrong therefore is the statement "Initial prime is". As the same program is used for all examples, perhaps what we could say is that the stated prime is part of a sequence with consecutive gaps of the stated size.

2022-01-19, 23:46   #19
Bobby Jacobs

May 2018

2·3·43 Posts

Quote:
Originally Posted by robert44444uk
Quote:
 Originally Posted by robert44444uk Here is an interesting pair of graphs - they shows he median gap of a range of all gaps for primes 2 to x, with x up to 999983. The second graph ignores the results up to x=4409 so as to see the fine detail of the later ranges. I plan to carry the range up to prime range from 2 to 1e7, although the work is relatively intensive on resources it does look as if the median gap is on the rise again at 1e6 and it is worth looking at this.
Please use the thread "Median Gaps" for further postings on this theme.
Did you tell yourself which thread to use? Weird.

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