20180623, 12:10  #1 
May 2018
2×7×17 Posts 
Prime gaps above 2^64
Will we be searching for prime gaps above 2^{64}?

20180626, 11:05  #2 
Jun 2003
Oxford, UK
5×397 Posts 
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. 
20180626, 22:37  #3 
May 2018
238_{10} Posts 
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 2^{64}.

20180627, 01:27  #4  
Jun 2015
Vallejo, CA/.
2^{2}×257 Posts 
Quote:
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) 

20180627, 02:47  #5 
Jun 2003
3·5·347 Posts 

20180627, 05:12  #6 
Jun 2015
Vallejo, CA/.
2004_{8} Posts 

20180627, 09:24  #7 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
13·457 Posts 
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 64bits? 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 20180627 at 09:25 
20180627, 09:39  #8  
Jun 2003
Oxford, UK
5·397 Posts 
Quote:


20180628, 00:38  #9 
May 2018
2·7·17 Posts 
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.

20180628, 10:41  #10  
Jun 2015
Vallejo, CA/.
2^{2}×257 Posts 
Quote:
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 

20180701, 18:39  #11 
May 2018
2·7·17 Posts 
I bet the next maximal prime gap after 2^{64} will also have record merit.

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