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.
|
|
|
|
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