2^9092392+40291 is a probable prime!

[philmoore]

Your estimate of 10000 is right. I'll be interested to hear what you find.

[TimSorbet]

First report:
I have searched 10000<=n<=10100. 27373 candidates after sieving to 1001M, expected primes 145.03, observed primes 168. This is 1.91 standard deviations above the expected amount (the std dev of a Poisson distribution is the square root of the expected, in this case 12.04). The chances of truly Poisson-distributed events falling within 1 standard deviation is 68.27%, within 2: 95.45%, within 3: 99.73%.
The expected primes for 10K-20K is 10020.86, making the standard deviation 100.10. I'd think there's something worth investigating if it's over +1.645 std devs at that point after taking away the 25 'extra' primes already observed, (sure, we observed 'extra' so far, but the expected primes remaining to be found is still about 10020.86-145.03=9875.83) which should only have a 10% chance of happening. To me, that would strongly suggest there's more here than random luck.

[philmoore]

I just re-did the calculations from the INTEGERS paper on the probability of successfully solving this problem, and using the weights I used then, I get that we had only a 0.55% chance of solving this problem by n=9092392. I later used some of our sieving data to revise our Proth weight data, but this would have only made our chances smaller! To solve a problem that we had only a 1 chance in 180 (or worse) of solving is truly remarkable. Maybe Tim's data can shed some light on the problem, but I suspect that we were also lucky. Maybe it is time to buy a lottery ticket!?

[TimSorbet]

Second milestone:
10K-11K done, expected 1380.39 primes, found 1401, +0.55 std devs. Counting only from 10.1K-11K, it's actually a bit lower than expected: -0.06 std devs. At this point, I'd guess that there's nothing making it higher than it should be, and that this project was just extremely lucky.

[philmoore]

Taking strong prp bases 47, 53, 59, and 61. (Only 67 and 71 are still available.)

[Jeff Gilchrist]

Originally Posted by philmoore

Taking strong prp bases 47, 53, 59, and 61. (Only 67 and 71 are still available.)

I will take 67 and 71 now. Here are most of my results so far:

11
SSE2 Proth FFT: size=(1048576,17.342)
Probable_prime_residueis_plus1

13
SSE2 Proth FFT: size=(1048576,17.342)
Probable_prime_residueis_minus1

17
SSE2 Proth FFT: size=(1048576,17.342)
Probable_prime_residueis_plus1

19
SSE2 Proth FFT: size=(1048576,17.342)
Probable_prime_residueis_minus1

31
SSE2 Proth FFT: size=(1048576,17.342)
Probable_prime_residueis_plus1

37
SSE2 Proth FFT: size=(1048576,17.342)
Probable_prime_residueis_plus1

41
SSE2 Proth FFT: size=(1048576,17.342)
Probable_prime_residueis_plus1

43
SSE2 Proth FFT: size=(1048576,17.342)
Probable_prime_residueis_minus1

The other two I should have in a day or so, and I will run these last two starting now.

Jeff.

[philmoore]

I have also confirmed bases 5 and 7, both with residues -1, and am now working on 47 and 53. We are 12 for 12, I don't think there is much doubt how the other eight tests will work out! Any progress, Justin, on your tests? Jeff and I have been corresponding about pfgw, we are running an old version which seems to be faster than the newer version, surprisingly. The new version appears to be using the GMP library rather than GWNUM, but even a 6-year-old GWNUM algorithm seems to beat GMP. That could explain your slow progress, Justin; if you decide to kill it, I don't blame you. At any rate, Jeff and I have generated some good suggestions for the next version of pfgw, including an option to do strong prp tests.

[enderak]

Progress is slow but steady. I guess because it's only running on 1 core and unlike using your script on different bases, I don't know how I would split it across cores. It says it's using GWNUM 26.4.

This is all that is in my log file:

Primality testing 2^9092392+40291 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N+1 test using discriminant 5, base 1+sqrt(5)

The N+1 test is only about 20% done, it's still going to be at least a week for that to finish, and then hopefully I'll have at least some sort of result.

[mart_r]

Originally Posted by philmoore

I just re-did the calculations from the INTEGERS paper on the probability of successfully solving this problem, and using the weights I used then, I get that we had only a 0.55% chance of solving this problem by n=9092392. I later used some of our sieving data to revise our Proth weight data, but this would have only made our chances smaller! To solve a problem that we had only a 1 chance in 180 (or worse) of solving is truly remarkable.

So that's where my luck in finding a top-10 world-record k-tuplet has gone to... now I know.

Anyway, congrats on this remarkable find.

[Jeff Gilchrist]

Two more finished.

23
SSE2 Proth FFT: size=(1048576,17.342)
Probable_prime_residueis_plus1

29
SSE2 Proth FFT: size=(1048576,17.342)
Probable_prime_residueis_plus1

[TimSorbet]

I finished my search, and found 9977 primes. With 10020.86 expected, this is -0.44 std devs from the expected. In my opinion, the results couldn't be clearer: on average, dual Sierpinski numbers are no more likely to be prime than they "should" be. Now, this doesn't entirely remove the possibility that (very-)low-weight k's produce more primes than they should or that the 5 k's this project search produce more primes than they should (the latter would be quite hard to prove with any certainty). I've attached the primes. If someone calculates the weight of each k I searched and groups low weight k's together, they might be able to see if they produce as much as expected, or more or less. But take note that low weight k's tend to scatter from their expected far more than high weight k's, (if you make a scatter plot of the weight of each k vs its ratio of expected to observed primes, you'll see a funnel-like shape with the low weight k's being the large, i.e. more scattered, end of it) so a few examples in one extreme or the other mean nothing.
I still have the sieved file if someone wants it for these weight calculations. I can upload it on request. (4.70 MB zipped) If nobody else seems interested, I might do it some time.