The probable primes

[engracio]

Well I am glad your wu is moving along. Mine after 10 days is back to test 10 again. It has been back and forth all the way up to test 16. So far backtrack count=20.

I'll give it another week, if I don't make any decent progress no mas.

[Cybertronic]

20 backtracks ? Wow!
Do you run on 4 cores ?

[philmoore]

I just received a remarkable email from Professor Wilfrid Keller in Hamburg:

February 17, 2010
Dear Phil Moore,

With reference to your list of PRPs at

http://www.mersenneforum.org/showpos...66&postcount=1

I wanted to report that 2^31483+29333 (9478 digits) and
2^56363+26213 (16967 digits) might be removed from the list,
as I found the smaller primes 2^1891+29333 (570 digits) and
2^1271+26213 (383 digits). Primality was established using
Marcel Martin's Primo and kindly verified by David Broadhurst:

> I confirm your findings, completely.
> Here are APR-CL proofs by Pari-GP:
>
> parisize = 400000000, primelimit = 20000000
> ? isprime(2^1271 + 26213)
> %1 = 1
> ? isprime(2^1891 + 29333)
> %2 = 1

Luckily, none of the above-mentioned PRPs had previously been
attacked with Primo!

Let me tell you the background of this. At the suggestion of John
Blazek from PrimeGrid, I recently reworked my web page

http://www.prothsearch.net/sierp.html

on Sierpinski's problem, which had been "frozen" in November 2002.
The desire was to "see all data in one central location", as John
put it.

In that context, David Broadhurst pointed me to your current work
on the "dual" Problem. I have to admit that I wasn't aware, at
that point, of the impressive four PRPs discovered within the
frame of "Five or Bust" (truly remarkable!). I only knew of the
two record Primo "certifications" by Norman Luhn, but probably
didn't relate them to the "dual problem".

Also, I hadn't looked at Payam Samidoost's page

http://sierpinski.insider.com/dual

for a long time.

Revisiting Payam's page, and after having "practised" with the
"prime Sierpinski problem", I was curious to compute the
frequencies similar to the f(m) for the original Sierpinski problem,
to get some insight into the different "elimination behaviour". For
instance, it was interesting to see that in the range in question
there are 6714 cases where k + 2^2 = (k + 2^1) + 2^1 = p (a prime)
with k + 2^1 composite, so that two consecutive odd k's are
eliminated at once in each case.

Continuing with my "reproduction of known results", I finally
discovered the two primes which in fact should have appeared on
the "list of all k < 78557 such that the first probable prime
k + 2^n found" has n within 1000 < n < 10000.

Everything else seems to have been verified (as far as PRPs were
concerned) up to n < 2^12 = 4096.

May I finally mention that I greatly enjoyed your paper on the
"mixed problem", and also your delightful talk on "Perfect, Prime,
and Sierpinski Numbers".

With kind regards,

Wilfrid Keller

We are so lucky that neither of the two probable primes in our list had been subjected to the torture test via Primo! I have now deleted them from post #1 in this thread. When I took up this project in 2007, I verified that all of the odd k values < 78557 not listed in Payam Samidoost's webpage had a prime k+2^n with n < 1000. But I assumed that the data up to n < 20,000, apparently discovered by Mark Rodenkirch and verified by David Broadhurst, was essentially accurate. Mark and David are very careful people, so I suspect a software problem with an old version of pfgw. But Professor Keller has only verified this data up to n = 4096, so it may be worthwhile to verify the rest of it. I currently have verification data in progress for the 7 sequences 37967, 60451, 75353, 28433, 8543, 2131, and 41693. If you care to verify that any of the remaining sequences in post #1 of this thread have no probable prime less than that listed, post below.

[Cybertronic]

I checked over that 2^ 38090 +47269 is the first prime of 2^n+47269.
So my work is not waste.

[philmoore]

In addition to the sequences represented by the 30 primes/probable primes now listed in post #1 of this thread, plus the 40291 sequence, there are an additional 16 sequences for which we should verify that the proven primes below are indeed the smallest in each sequence. I am reasonably certain about the eight sequences I started with in 2007, including the five of Five or Bust, as I started my search at n = 1, and double-checking has so far confirmed that at least the early tests were accurate. Wilfrid Keller has checked all of these sequences up to n = 4096.

2⁴⁸⁷⁰+20209
2⁵³³⁵+41453
2⁵⁷⁵⁹+64643
2⁵⁷⁶⁰+5101
2⁵⁸⁸³+24953
2⁶⁰²²+48859
2⁶¹⁴⁴+26491
2⁶²⁶²+49279
2⁶⁴⁷⁷+56717
2⁶⁴⁹⁶+31111
2⁶⁶⁴⁹+6887
2⁹⁶⁹⁶+48091
2¹¹¹⁵²+23971
2¹²⁰⁷⁵+14033
2¹²⁷¹⁵+14573
2¹⁶³⁸⁹+67607

(Note that 2⁶⁰²²+48859 has two digits transposed on Payam Samidoost's page.)

If no one else does it first, I will check as many as I can the weekend of 27 February. Of course, double-checking the last few sequences is an ongoing concern, but I should be able to confirm the rest, and at least verify whether there are any more errors on Samidoost's page.

[Mini-Geek]

Originally Posted by philmoore

Wilfrid Keller has checked all of these sequences up to n = 4096.

2⁴⁸⁷⁰+20209
2⁵³³⁵+41453
2⁵⁷⁵⁹+64643
2⁵⁷⁶⁰+5101
2⁵⁸⁸³+24953
2⁶⁰²²+48859
2⁶¹⁴⁴+26491
2⁶²⁶²+49279
2⁶⁴⁷⁷+56717
2⁶⁴⁹⁶+31111
2⁶⁶⁴⁹+6887
2⁹⁶⁹⁶+48091
2¹¹¹⁵²+23971
2¹²⁰⁷⁵+14033
2¹²⁷¹⁵+14573
2¹⁶³⁸⁹+67607

I have just verified these from n=4000 and found the first PRPs at the numbers listed on all of them (i.e. assuming Wilfrid Keller and I did not make any big mistakes, the n's listed are indeed the lowest PRPs for those k's).

[philmoore]

Originally Posted by Mini-Geek

I have just verified these from n=4000 and found the first PRPs at the numbers listed on all of them (i.e. assuming Wilfrid Keller and I did not make any big mistakes, the n's listed are indeed the lowest PRPs for those k's).

Thanks, Tim. What software did you use for the verification?

We still have the primes/probable primes listed in post #1:

2²¹⁹⁵⁴+77899
2²²⁴⁶⁴+63691
2²⁴⁹¹⁰+62029
2²⁵⁵⁶³+22193
2²⁶⁷⁹⁵+57083
2²⁶⁸²⁷+77783
2²⁸⁹⁷⁸+34429
2²⁹⁷²⁷+20273
2³¹⁵⁴⁴+19081
2³³⁵⁴⁸+4471
2³⁸⁰⁹⁰+47269 (checked by Cybertronic)
2⁵⁶³⁶⁶+39079
2⁶¹⁷⁹²+21661
2⁷³³⁶⁰+10711
2⁷³⁸⁴⁵+14717
2¹⁰³⁷⁶⁶+17659
2¹⁰⁴⁰⁹⁵+7013
2¹⁰⁵⁷⁸⁹+48527
2¹³⁹⁹⁶⁴+35461
2¹⁴⁸²²⁷+60443
2¹⁷⁶¹⁷⁷+60947
2³⁰⁴⁰¹⁵+64133
2³⁰⁸⁸⁰⁹+37967
2⁵⁵¹⁵⁴²+19249
2⁹⁸³⁶²⁰+60451
2^1191375+8543
2^1518191+75353
2^2249255+28433
2^4583176+2131
2^5146295+41693

It might be especially helpful to do 2³¹⁵⁴⁴+19081, as Engracio is not too far into the ECCP proof yet, and also 2²⁸⁹⁷⁸+34429, as it is probably the next one on the list to prove.

[Mini-Geek]

Originally Posted by philmoore

Thanks, Tim. What software did you use for the verification?

PFGW with the -f option and an ABC2 file (not a pre-sieved sort of file). Was pretty fast, but I wouldn't want to check all the larger ones that way. Pre-sieving a file would definitely help for that.

[philmoore]

I presieved the sequence 2ⁿ+19081 with srsieve, and got down to 130 tests or so, then confirmed that Engracio's number 2³¹⁵⁴⁴+19081 was indeed the smallest in that sequence. I will do the rest soon, but I just didn't want him to continue running Primo on it if it was not the smallest.

How is it going, Engracio? Any progress?

[engracio]

Originally Posted by philmoore

I presieved the sequence 2ⁿ+19081 with srsieve, and got down to 130 tests or so, then confirmed that Engracio's number 2³¹⁵⁴⁴+19081 was indeed the smallest in that sequence. I will do the rest soon, but I just didn't want him to continue running Primo on it if it was not the smallest.

How is it going, Engracio? Any progress?

Thanks Phil. It is very slow. I have not given up yet but if I backtracked more than moving forward. I might consider prp'ng more productive.

[Cybertronic]

Originally Posted by engracio

I have not given up yet but if I backtracked more than moving forward. I might consider prp'ng more productive.

Indeed Engracio, I can sing a song about this......and now add
2000 decimal digits to your number :surprised

I believe 11467 digtis with PRIMO is the highest of emotions.