The dual Sierpinski problem search
Now that we know 78557+2^n is always composite, we can define a project similar to Wilfrid Keller and Ray Ballinger's search for the numbers of the form k+2^n. That is we are trying to find a prime in each remaining sequence of integers of the form k+2^n (fixed k) for which no prime is found yet. Please contact me to save your reservations and results under your name.
New's Flash
On August 17, 2002 Payam Samidoost found the 166031 digit PRP
19249+2^551542
This is the
greatest PRP
as well as the greatest dual Proth
known.
Now there remains only 8 candidates for the dual of Sierpinski problem.
Also
This number removes one of the two remaining mixed Sierpinski problem candidates.
Now there remains only one candidate to remove, namely the k=28433,
to solve the Sierpinski problem.
The list of all k<78,557 such that k+2^n is composite for each n<100,000
| k | n | reserved by | last update |
dual n |
ProthWeight | |
| 2131 | 453,000 | Payam Samidoost | July 20, 2002 | 44 | 0.08450 | |
| 7013 | 104,095 | David Broadhurst | May 29, 2002 | 126,113 | 0.04682 | |
| 8543 | 284,000 | Martin Schroeder | September 1, 2002 | 5,793 | 0.06337 | |
| 17659 | 103,766 | David Broadhurst | May 28, 2002 | 34 | 0.12047 | |
| 19249 | 551,542 | Payam Samidoost | August 17, 2002 | [1,055,000] | 0.04339 | |
| 28433 | 400,000 | Payam Samidoost | September 3, 2002 | [1,190,000] | 0.05424 | |
| 35461 | 139,964 | Marcin Lipinski | May 31, 2002 | 4 | 0.11247 | |
| 37967 | 230,000 | Marcin Lipinski | July 3, 2002 | 23 | 0.15015 | |
| 40291 | 269,000 | Richard Heylen | August 5, 2002 | 8 | 0.09649 | |
| 41693 | 486,000 | Michael Porter | October 1, 2002 | 33 | 0.09135 | |
| 48527 | 105,789 | David Broadhurst | May 28, 2002 | 951 | 0.09877 | |
| 60443 | 148,227 | David Broadhurst | May 28, 2002 | 95,901 | 0.06509 | |
| 60451 | 600,000 | free | 44 | 0.16386 | ||
| 60947 | 176,177 | David Broadhurst | May 25, 2002 | 783 | 0.16214 | |
| 64133 | 304,015 | David Broadhurst | June 4, 2002 | 161 | 0.17870 | |
| 75353 | 600,000 | free | 1 | 0.08735 |
The dual n indicates the smallest n such that k*2^n+1 is prime.
2131 [100-170 Broadhurst 30/05/2002]
8543 [100-170 Broadhurst 30/05/2002]
19249 [100-250 Broadhurst 30/05/2002]
28433 [100-250 Broadhurst 30/05/2002]
40291 [100-170 Broadhurst 31/05/2002]
41693 [100-170 Broadhurst 31/05/2002]
60641 [100-600 Broadhurst 27/06/2002]
75353 [100-600 Broadhurst 03/06/2002]
Comparison with the old days of Sierpinski search shows a major difference between the count of the remaining candidates. THEY ARE FAR MORE RARE IN THE CASE OF DUALS. (good news for the dual project)
The reason:
Note that
every odd integer has a unique representation in Proth form, BUT NOT IN ITS DUAL FORM.
(Except 2^n+1 which are their self duals)
for example:
31 = 15+2^4 = 23+2^3 = 27+2^2 = 29+2^1
Most of the Sierpinski or dual Sierpinski candidate sequences are removed by their small prime members. Since each small prime have more than one representation in dual form, the dual candidates are more likely to be removed.
The smallest dual Sierpinski candidate
| k | n |
| 3 | 1 |
| 7 | 2 |
| 23 | 3 |
| 31 | 4 |
| 47 | 5 |
| 61 | 8 |
| 139 | 10 |
| 271 | 20 |
| 287 | 29 |
| 773 | 955 |
| 2131 | [453,000] |
The list of all k<78,557 such that the first prime of the form k+2^n is found within 10,000
The gray numbers are the
ProthWeights
The removed candidates with their primes are in green
[trial division limits are written in black]
Mark Rodenkirch [May 17, 2002] tested all the remaining candidates up to 20,000.
David Broadhurst [May 30, 2002] verified all the results up to 100,000.
For the results with n>100,000 please see
above
0.084 2131 [100000 Samidoost]
0.184 4471 33548 Lipinski [May 17, 2002]
0.046 7013 [100000 Broadhurst]
0.063 8543 [100000 Broadhurst]
0.176
10711 73360 Broadhurst [May 20, 2002]
0.094
14033 12075 Samidoost, Rodenkirch [May 17, 2002]
0.244
14573 12715 Rodenkirch [May 17, 2002]
0.188
14717 73845 Broadhurst [May 20, 2002]
0.120
17659 [100000 Broadhurst]
0.102
19081 31544 Broadhurst [May 20, 2002]
0.043
19249
[63000 Samidoost][100000 Broadhurst]
0.196
20273 29727 Broadhurst [May 20, 2002]
0.309
21661 61792 Broadhurst [May 20, 2002]
0.066
22193 25563 Hoogendoorn [May 17, 2002]
0.062
23971 11152 Rodenkirch [May 17, 2002]
0.232
26213 56363 Broadhurst [May 20, 2002]
0.054
28433
[64000 Samidoost][100000 Broadhurst]
0.070
29333 31483 Hoogendoorn, Lipinski [May 17, 2002]
0.223
34429 28978 Lipinski [May 19, 2002]
0.112
35461 [100000 Lipinski]
0.150
37967 [100000 Lipinski]
0.040
39079 56366 Lipinski [May 24, 2002]
0.096
40291 [38000 Hoogendoorn][100000 Lipinski 25-05-2002]
0.091
41693 [100000 Lipinski 27-05-2002]
0.164
47269 38090 Broadhurst [May 20, 2002]
0.098
48527 [100000] Broadhurst
0.173
57083 26795 Broadhurst [May 20, 2002]
0.065
60443 [90000 Rodenkirch][100000 Broadhurst]
0.163
60451 [43000 Hoogendoorn][100000 Broadhurst]
0.162
60947 [100000 Broadhurst]
0.203
62029 24910, 29550 Broadhurst [May 20, 2002]
0.198
63691 22464 Broadhurst [May 20, 2002]
0.178
64133 [90000 Rodenkirch][100000 Broadhurst]
0.035
67607
16389 Fougeron [May 14, 2002] 46549 Samidoost [Nov 14, 2001][72000]
0.087
75353 [100000 Broadhurst]
0.149
77783 26827 Broadhurst [May 17, 2002]
0.102
77899 21954 Broadhurst [May 17, 2002]
The list of all k<78,557 such that the first probable prime in k+2^n found within 1000
Thanks to Mark Rodenkirch who had found the following probable primes [May 16, 2002] all the remaining candidates are clean up to 10000. Special thanks to David Broadhurst who had verified the whole range [May 23, 2002] and found the missed number 29777+2^1885 which is written in red. Unfortunately Marcin Lipinski which by bad chance had focused just over this candidate had tested it by trial division up to 105000.
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Known Probable Primes of the form k+2^n, n>50000, k<2^n
| rank | k | n | who | date |
| 1 | 19249 | 551542 | Payam Samidoost | August 17, 2002 |
| 2 | 64133 | 304015 | David Broadhurst | June 4, 2002 |
| 3 | 204129 | 204129 | Henri Lifchitz | 04/2002 |
| 4 | 41877 | 180001 | Jim Fougeron | 08/2002 |
| 5 | 60947 | 176177 | David Broadhurst | May 25, 2002 |
| 6 | 60443 | 148227 | David Broadhurst | May 28, 2002 |
| 7 | 35461 | 139964 | Marcin Lipinski | May 31, 2002 |
| 8 | 49653 | 131072 | Henri Lifchitz | 04/2002 |
| 9 | 5851 | 131072 | Henri Lifchitz | 09/2001 |
| 10 | 123771 | 123773 | Renauld Lifchitz | 07/2002 |
| 11 | 3 | 122550 | Mike Oakes | 07/2001 |
| 12 | 48527 | 105789 | David Broadhurst | May 28, 2002 |
| 13 | 7013 | 104095 | David Broadhurst | May 29, 2002 |
| 14 | 17659 | 103766 | David Broadhurst | May 28, 2002 |
| 15 | 99069^2 | 99069 | Rob Binnekamp | 06/2001 |
| 16 | 88071 | 88071 | Henri Lifchitz | 09/2001 |
| 17 | 14287 | 83500 | William Garnett | 02/2002 |
| 18 | 9 | 80949 | Mike Oakes | 08/2001 |
| 19 | 75765 | 75764 | Henri Lifchitz | 12/2001 |
| 20 | 14717 | 73845 | David Broadhurst | May 20, 2002 |
| 21 | 10711 | 73360 | David Broadhurst | May 20, 2002 |
| 22 | 21661 | 61792 | David Broadhurst | May 20, 2002 |
| 23 | 29705 | 60023 | Milton Brown | 05/2001 |
| 24 | 3 | 58312 | Mike Oakes | 07/2001 |
| 25 | 57285^2 | 57285 | Rob Binnekamp | 06/2001 |
| 26 | 39079 | 56366 | Marcin Lipinski | May 24, 2002 |
| 27 | 26213 | 56363 | David Broadhurst | May 20, 2002 |
| 28 | 3 | 55456 | Mike Oakes | 07/2001 |
| 29 | 9 | 50335 | Mike Oakes | 08/2001 |
| 30 | 25215 | 50000 | Milton Brown | 05/2001 |
If you know more probable primes of the form k+2^n (n>=50000, k<2^n)
contact me
please.
You can find more PRP's in
Henri Lifchitz's top 1000 probable primes list
.
This page is maintained by Payam Samidoost
Last updated: October 2, 2002
