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.

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]
0.176 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]
0.102 19081 31544 Broadhurst [May 20, 2002]
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]
0.070 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]
0.173 57083 26795 Broadhurst [May 20, 2002]
0.203 62029 24910, 29550 Broadhurst [May 20, 2002]
0.198 63691 22464 Broadhurst [May 20, 2002]
0.035 67607 16389 Fougeron [May 14, 2002] 46549 Samidoost [Nov 14, 2001][72000]
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.

 2491 3536 5101 5760 6379 1338 6887 6649 8447 2997 9833 2219 14551 1548 15623 3719 16519 1126 18527 1709 20209 4870 21143 1927 23147 1197 23221 1668

 24953 5883 26491 6144 29777 1885 30197 2153 31111 6496 31369 2322 31951 1404 32449 1814 32513 1295 36083 3447 36721 2432 37212 2121 38387 1445 40351 1036

 40613 1087 41453 5335 43579 1190 48091 9696 48331 3564 48589 6022 48961 2424 49279 6262 49577 2461 50839 1974 52339 2294 53119 1210 53359 2694 56717 6477

 59071 1760 60961 1356 64643 5759 65033 1147 65089 1038 65719 1954 69593 3415 69709 1770 70321 2500 72679 1818 73373 2531 73583 1903 75841 1180 77041 2692

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 .