List of most small twins of form k*2^n+/-1

[gd_barnes]

Originally Posted by kar_bon

next update is online!
a second graph including all twins for 10,000<n<32,200 and k<100,000 found by Gary Barnes!
the given forumla is not quite correct because of the missing twins above k=100,000!

Would anyone like to fill in k=100K-1M for n<10K ? It would not be a big effort CPU-wise but listing them on web pages would be quite a headache, which is why I opted not to do it. It would best be shown on 10-15 web pages so that any one page is not too large.


Gary

[robert44444uk]

Gary

The suggestion to look at 70000-71000 is not supposed to take away from what you are doing, I believe that the concept of "no prime left behind" is truly valid and the list of primes that you have already found and archived is exceptional. You should not give up this exercise.

The purpose of the lowest k search has been to prove a point, and I don't see any real advantage of taking that exercise any further, most things that might be derived from it have been derived except for the following:

1. is there an upper limit for the first instance k that is related to the n level being searched? I noticed that in the first 6000 n, there was ALWAYS at least 1 k<80M that provided a twin, and that there were no rogue ks that were greater than 10ln(n) - (I think that such rogue values exist by the way).

2. the comparison between medians of 50, consecutive n, first k values and X=0.24 n^2 appear to produce close to 100% of values between 50% and 200% of X. Smoothed medians appear to be bounded with respect to n^2. Only more n values tested will provide confirmation of this fact.

In terms of DC efforts, my understanding is that the current suggested test at 70-71K is too small for automated DC, but that a good result (i.e. twin prime found) will set the ground for a DC exercise at a suitable n level such that a positive result (largest twin) can be achieved. These days the total amount of DC capacity is less than it has been in the past, if we access this treasure then we have to ensure success in a reasonable time period!!!! The level of interest in 333333 is dropping off, and no twin found.

So the n=70-71K test would not be automated DC, but could be achieved with a modest group of resources, perhaps working off a large file with all 1000n pre sieved to 120 billion. The file will contain approx 1.5 million values, and prp testing will need to be shared around to complete the exercise quickly. In parallel, setting up the automated DC effort can be carried out.

[gd_barnes]

Originally Posted by robert44444uk

Gary

The suggestion to look at 70000-71000 is not supposed to take away from what you are doing, I believe that the concept of "no prime left behind" is truly valid and the list of primes that you have already found and archived is exceptional. You should not give up this exercise.

The purpose of the lowest k search has been to prove a point, and I don't see any real advantage of taking that exercise any further, most things that might be derived from it have been derived except for the following:

1. is there an upper limit for the first instance k that is related to the n level being searched? I noticed that in the first 6000 n, there was ALWAYS at least 1 k<80M that provided a twin, and that there were no rogue ks that were greater than 10ln(n) - (I think that such rogue values exist by the way).

2. the comparison between medians of 50, consecutive n, first k values and X=0.24 n^2 appear to produce close to 100% of values between 50% and 200% of X. Smoothed medians appear to be bounded with respect to n^2. Only more n values tested will provide confirmation of this fact.

In terms of DC efforts, my understanding is that the current suggested test at 70-71K is too small for automated DC, but that a good result (i.e. twin prime found) will set the ground for a DC exercise at a suitable n level such that a positive result (largest twin) can be achieved. These days the total amount of DC capacity is less than it has been in the past, if we access this treasure then we have to ensure success in a reasonable time period!!!! The level of interest in 333333 is dropping off, and no twin found.

So the n=70-71K test would not be automated DC, but could be achieved with a modest group of resources, perhaps working off a large file with all 1000n pre sieved to 120 billion. The file will contain approx 1.5 million values, and prp testing will need to be shared around to complete the exercise quickly. In parallel, setting up the automated DC effort can be carried out.

I agree that n=70K-71K is too small for a DC project. But what I meant was a DC project for the effort that I am doing for n=36.2K-100K (or higher) for k=3-1M. It's just an idea that I threw out.

Regardless, my 'all twin' effort will start again by the end of this weekend. I'll have 1 core and a little bit later on...2 cores of one of my new quads running on it.


Gary

[roger]

It looks like I'm around 3/4 done my range. Sorry it's taken so long, my computer has been a bit odd, insisting that it can only use 800Mhz when I have a 1.7Ghz processor. It's done this before, and I really have no clue as to why...

I would like to reserve a range once this one is done, but I'll wait until then to see what's appealing. This time, I'll have the correct sieve depths and parameters so it should take much less time

[Cruelty]

Originally Posted by roger

It looks like I'm around 3/4 done my range. Sorry it's taken so long, my computer has been a bit odd, insisting that it can only use 800Mhz when I have a 1.7Ghz processor. It's done this before, and I really have no clue as to why...

sounds like some cool'n'quiet or speedstep feature is enabled - have you tried to disable it?
It can also mean CPU throttling...

[roger]

Is this 'cool'n'quiet or speedstep' a feature on LLR, or my computer (Dell)? I haven't tried to disable it because I've never heard of it but how would I do it if that's what's happening?

So far, I've only tried restarting the computer, and that usually works (though sometimes I have to do it twice...

Also, what's throttling? Thanks!

[roger]

Another quirk:

Even though -oStopOnSuccess=1 is included in the .bat file, the program will test all candidates before starting the next n-value...

EDIT: and now it's working again...

[gd_barnes]

An update on my effort for the "all" twin search for k<1M and n<100K:

Before I left on a business trip 11 days ago, I put a full quad on the range of n=36.2K-40K, n=800 on one core and n=1K on the other 3, to get this effort moving again. n=36.2K-36.9K, 37K-37.7K, 38K-38.7K, and 39K-39.7K have completed so far.

7 more twins have been found; 1 for k<100K. Once all of the ranges have completed, I'll update my web page and do the same for n=40K-44K. ETA for completion to n=40K is now ~4 days.

I'll proceed that way up until n=48K and then halve the range sizes per core.

[amphoria]

Originally Posted by robert44444uk

I strongly advise a test at lower than 200000 to check if this approach is worthwhile. The test should be at greater than the range Gary is currently checking, say n range 40000-41000, testing up to k=670000, to see if this produces at least one twin. The alternative test could be at nrange=67000-68000. which would be archivable, testing to 1860000.

The twin prime that I found recently was as a result of testing n=67000-68000 with k up to 1860000, which gives further support to this approach. In the end I only searched about 1/3 of the range before finding the twin as I was using 2 cores.

Dave

[roger]

Has anyone reserved the range of 6080->8825? I think I'll take 6080->7000 if they're available, and do a bit in the 40,000 and 50,000 range (maybe to 10M or so for a couple n's for a semireasonable chance of a twin ).

97% done 8825->9500! Looks like the average k-value is going up quite quickly... there are a couple of n's without twins below 120M already!

[robert44444uk]

Roger this range (6080-8825) is now free for you.