190927*2^72289-1 is prime, written earlier.
173587*2^172609-1 is prime.
112391 reserved by me now.

112391*2^159730-1 is prime! Time : 40.493 sec.

I'm reserving 239107. All 4 remaining k's are now reserved.

update
 

17861 (17861*2^98954-1 is prime! Time : 23.000 sec. by jasong)
23651 (23651*2^237506-1 is prime! Time : 167.859 sec.) found by jasong
77167 (77167*2^153441-1 is prime! Time : 34.146 sec.) found by jasong
170467 (170467*2^55273-1 is prime. by Jens K Andersen)
173587 (173587*2^172609-1 is prime.) found by Jens K Andersen
175567 reserved by jasong
190927 (190927*2^72289-1 is prime.) found by Jens K Andersen
112391 112391 reserved by Jens K Andersen
239107 reserved by Jens K Andersen.

testing begins at n=85,000 and continues to n=500,000.

Great!
I wrote earlier that 112391*2^159730-1 is prime so we are down to 2 k's. Unfortunately one of them is the expected hardest: 239107 which has quickly growing candidates and is currently tested to 244000. I estimate less than 50% chance it has a prime below 1,000,000 digits.
175567 looks much more promising.

[QUOTE=Jens K Andersen;116351]Great!
I wrote earlier that 112391*2^159730-1 is prime so we are down to 2 k's. Unfortunately one of them is the expected hardest: 239107 which has quickly growing candidates and is currently tested to 244000. I estimate less than 50% chance it has a prime below 1,000,000 digits.
175567 looks much more promising.[/QUOTE]
Only 2 to go. That's fantastic.

I like to jump from project to project, and because I'm very impulsive, I reserved work in another project before I remembered my k. 175567*2^187425-1 is as far as I got. If anybody wants it, I'm unreserving it.

This project looks like it might be completed fairly quickly. Then, again, one or both of the remaining ks may prove to be very, very stubborn.

239107 gave no prime to 500000.
Reserving 175567 from 187425 to 500000.

If anybody decides to sieve above n=500,000 please both check here to see if anybody else has beaten you to it, and be sure to post your intentions either before you start, or at least within hours of starting.

175567 gave no prime to 500000.
175567 and 239107 have not been sieved above n=500,000.
I have stopped working on this project.
The following is a summary of the search so far.

Consider k values for which k*2^n-1 is composite for all n>0.
Odd such k are called Riesel numbers.
The goal of the Riesel problem is to find the smallest Riesel number.
The goal of our project is to prove that there is no even k below the smallest Riesel number, whatever it is. The smallest known is 509203.

The following PARI/GP script identifies odd k values which have an even multiple of form k*2^m that is a potential solution below the smallest Riesel number.
It first eliminates odd k which have no prime of form k*2^n-1 below 509203, because if any k*2^m below 509203 for such a k is a solution then k would itself be a Riesel number.
Second it eliminates k which give a prime between 509203 and k*2^1000-1.
The 61 remaining k values are printed.

? R=509203;L=1000;
? forstep(k=1,R,2,c=0;n=1;\
while(k*2^n<R,c+=isprime(k*2^n-1);n++);if(c,\
while(n<=L && !isprime(k*2^n-1),n++);\
if(n>L,print1(k", "))))
37, 337, 1589, 1721, 1807, 2257, 2317, 2683, 3775, 5857, 6869, 10021, 11887,
12401, 17861, 18089, 23651, 24161, 31453, 31841, 32257, 33373, 39817, 43151,
46411, 47653, 55687, 58331, 63367, 67001, 74857, 77167, 79601, 80771, 88115,
90907, 112391, 114367, 115451, 116257, 118447, 120457, 120997, 121061,
122017, 135787, 170467, 173467, 173587, 175567, 179677, 185347, 190357,
190927, 207397, 209737, 230407, 230827, 233221, 239107, 246787,

A prime has been found for 59 of the k values:
37*2^2553-1
337*2^11677-1
1589*2^1620-1
1721*2^1034-1
1807*2^1369-1
2257*2^1297-1
2317*2^2805-1
2683*2^2239-1
3775*2^1297-1
5857*2^4973-1
6869*2^45084-1
10021*2^1835-1
11887*2^1189-1
12401*2^26522-1
17861*2^98954-1
18089*2^1124-1
23651*2^237506-1
24161*2^8570-1
31453*2^1371-1
31841*2^1010-1
32257*2^1985-1
33373*2^5283-1
39817*2^1801-1
43151*2^23286-1
46411*2^2027-1
47653*2^1083-1
55687*2^1597-1
58331*2^1506-1
63367*2^1129-1
67001*2^9506-1
74857*2^1121-1
77167*2^153441-1
79601*2^3542-1
80771*2^9482-1
88115*2^2468-1
90907*2^4689-1
112391*2^159730-1
114367*2^1681-1
115451*2^6218-1
116257*2^1045-1
118447*2^14473-1
120457*2^1261-1
120997*2^2121-1
121061*2^2338-1
122017*2^1257-1
135787*2^7721-1
170467*2^55273-1
173467*2^6925-1
173587*2^172609-1
179677*2^2729-1
185347*2^1189-1
190357*2^15465-1
190927*2^72289-1
207397*2^5609-1
209737*2^1313-1
230407*2^1105-1
230827*2^4177-1
233221*2^1021-1
246787*2^1081-1

Used programs: PARI/GP, PrimeForm/GW, srsieve, LLR.
jasong found 17861*2^98954-1, 23651*2^237506-1, 77167*2^153441-1.
The largest found prime is 23651*2^237506-1.

Reserving k=239107 and 175567 for sieving
 

Reserving 239107 and 175567 for sieving.

I'll be sieving for n=500K-2,500K.

I'll sieve each to a billion. Then, as soon as I figure out how to combine the files, I'll use sr2sieve to sieve higher.

I'm unreserving the numbers and posting the sieve file.

[B]Edit by Max (8/30/09): Cleaned up attachment as the sieve file is now available on the [url=http://www.noprimeleftbehind.net/crus/]Conjectures 'R Us web site[/url]. (See the "Riesel Conjecture Reservations" page, under base 2.)[/B]

i inserted all k's mentioned here with their available data in the data-pages of [url]www.15k.org[/url]. the next update of these pages will be end november and then all k's are available there. so if anyone has some more infos for me (higher n values search -> more primes) i can insert them too. perhaps i push some k to higher n.