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 [100170 Broadhurst 30/05/2002]
8543 [100170
Broadhurst 30/05/2002]
19249 [100250 Broadhurst 30/05/2002]
28433
[100250 Broadhurst 30/05/2002]
40291 [100170 Broadhurst
31/05/2002]
41693 [100170 Broadhurst 31/05/2002]
60641 [100600
Broadhurst 27/06/2002]
75353 [100600 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 25052002]
0.091
41693 [100000 Lipinski 27052002]
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