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Old 2018-06-23, 12:10   #1
Bobby Jacobs
 
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Default Prime gaps above 2^64

Will we be searching for prime gaps above 264?
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Old 2018-06-26, 11:05   #2
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Originally Posted by Bobby Jacobs View Post
Will we be searching for prime gaps above 264?
I presume you are talking about continuing this comprehensive search. (There is an infinite number of primes larger than 2^64, and there are hence infinitely many gaps to look for).

Assuming the former - I think it is likely that the search will continue some stage, but there is no software written at present that operates at the levels of performance that Robert Gerbicz's gap12 program achieves. Hence there may be a lull in proceedings.
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Old 2018-06-26, 22:37   #3
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Yes. I am talking about the search. I would love to continue this search. It would be great to find out the next few maximal prime gaps after 264.
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Old 2018-06-27, 01:27   #4
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Quote:
Originally Posted by Bobby Jacobs View Post
Yes. I am talking about the search. I would love to continue this search. It would be great to find out the next few maximal prime gaps after 264.
A few interesting things would probably come IFF we would be able to find an efficient method of searching over 2^64
One of those would be to find a First Occurrence for Gap 1432. If this were the case all Gaps ≤ 1442 will be first occurrences (CFCs). This could presumably help in the searches beyond.

We might be able (this is a bit more complicated) to get a new maximal gap with merit ≥35.31 which would be the highest merit within our exhaustive search limits. For instance if it were found a gap of merit 35.50 at 1.85e19 it would be a gap ≥1476. If we found it at 2.0e19 it would have to be ≥1578 . I would be happy if we could find a new maximal gap (after 1530 and 1550 regardless of Merit)
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Old 2018-06-27, 02:47   #5
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For instance if it were found a gap of merit 35.50 at 1.85e19 it would be a gap ≥1476. If we found it at 2.0e19 it would have to be ≥1578 .
That 1476 should be a 1576.
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Old 2018-06-27, 05:12   #6
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That 1476 should be a 1576.
You are right! (1476 is already a CFC since 2009)
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Old 2018-06-27, 09:24   #7
henryzz
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The bit that slows down particularly is the prp test. This is slowed down by the modular exponentiation.

How many cpu cycles does one prp test take at 64-bits? What is the best we can do for 65?

I would suggest that a thread that contains code for a variety of different methods for 65+ bit modular exponentiation could be useful for the forum in general even if we decide it slows this project down too much.

Last fiddled with by henryzz on 2018-06-27 at 09:25
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Old 2018-06-27, 09:39   #8
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Quote:
Originally Posted by henryzz View Post
The bit that slows down particularly is the prp test. This is slowed down by the modular exponentiation.

How many cpu cycles does one prp test take at 64-bits? What is the best we can do for 65?

I would suggest that a thread that contains code for a variety of different methods for 65+ bit modular exponentiation could be useful for the forum in general even if we decide it slows this project down too much.
Good idea, henryzz. I'll open one up
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Old 2018-06-28, 00:38   #9
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I am surprised that we have not found a maximal prime gap with greater merit than the gap of size 1476. That gap has merit 35.3, but all of the known maximal gaps bigger than 1476 have merit under 35. There must be a gap with greater merit soon.
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Old 2018-06-28, 10:41   #10
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Quote:
Originally Posted by Bobby Jacobs View Post
I am surprised that we have not found a maximal prime gap with greater merit tha the gap of size 1476. That gap has merit 35.3, but all of the known maximal gaps bigger than 1476 have merit under 35. There must be a gap with greater merit soon.
As Tom Jones would say. It]s Not Unusual

There are 78 Maximal Gaps. We can include the "soon to be" gaps of 1530 and 1550 which will have merits smaller than 35.31, So, each of the 5 gaps after 1476 have smaller merits.

It happens 38 times.

Code:
No.     GAP
39	456	P11 = 25056082087	+19.04	Richard P. Brent	1973
40	464	P11 = 42652618343	18.96	Richard P. Brent	1973
41	468	P12 = 127976334671	18.30	Richard P. Brent	1973
42	474	P12 = 182226896239	18.28	Richard P. Brent	1973
43	486	P12 = 241160624143	18.54	Richard P. Brent	1973
44	490	P12 = 297501075799	18.55	Richard P. Brent	1973
45	500	P12 = 303371455241	18.91	Richard P. Brent	1973

Code:
No.     GAP
64	1132	P16 = 1693182318746371	       +32.28	Bertil Nyman	1999
65	1184	P17 = 43841547845541059	        30.90	Bertil Nyman	2002
66	1198	P17 = 55350776431903243	        31.07	Tomás Oliveira e Silva	2002
67	1220	P17 = 80873624627234849	        31.34	Tomás Oliveira e Silva	2003
68	1224	P18 = 203986478517455989	30.71	Tomás Oliveira e Silva	2005
69	1248	P18 = 218034721194214273	31.26	Tomás Oliveira e Silva	2005
70	1272	P18 = 305405826521087869	31.59	Tomás Oliveira e Silva	2006
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Old 2018-07-01, 18:39   #11
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I bet the next maximal prime gap after 264 will also have record merit.
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