The Riesel-Problem

Latest news:

2017-12-13: Riesel Prime 273809·2^8932416-1 found by W.Schwieger, PrimeGrid
2014-10-04: Riesel Prime 502573·2^7181987-1 found by D.Iakovlev, PrimeGrid
2014-10-02: Riesel Prime 402539·2^7173024-1 found by W.Darimont, PrimeGrid
2013-12-27: Riesel Prime 40597·2^6808509-1 found by F.Meador, PrimeGrid
2013-10-10: Riesel Prime 304207·2^6643565-1 found by R.Ready, PrimeGrid
2013-10-05: Riesel Prime 398023·2^6418059-1 found by V.Volynsky, PrimeGrid
2012-06-23: Riesel Prime 252191·2^5497878-1 found by J.Haller, PrimeGrid
2012-02-02: Riesel Prime 162941·2⁹⁹³⁷¹⁸-1 found by D.Domanov, PrimeGrid
2011-05-31: Riesel Prime 353159·2^4331116-1 found by J.Reinman, PrimeGrid
2011-05-26: Riesel Prime 141941·2^4299438-1 found by J.S.Brown, PrimeGrid
2011-05-08: Riesel Prime 415267·2^3771929-1 found by A.Tarasov, PrimeGrid
2011-05-08: Riesel Prime 123547·2^3804809-1 found by J.Luszczek, PrimeGrid
2011-04-05: Riesel Prime 65531·2^3629342-1 found by A.Schori, PrimeGrid
2011-01-14: Riesel Prime 428639·2^3506452-1 found by B.Melvold, PrimeGrid
2010-11-21: Riesel Prime 191249·2^3417696-1 found by J.Pritchard, PrimeGrid
2008-07-14: Error for k=103811 (n=111043 was given instead of 114034) found by W.Siemelink.
2008-06-24: Riesel Prime 485767·2^3609357-1 found by C.Cardall, RieselSieve
2008-05-01: Riesel Prime 113983·2^3201175-1 found I.Keogh, RieselSieve

Special thanks to W.Keller for his data and corrections. See also his page here.
The PrimeGrid Status page for their current search can be found here.

Definition :

Theorem: There exist infinitely many odd integers k such that k · 2ⁿ-1 is composite for every n ≥ 1.

Hans Riesel proved this in 1956 and showed that k₀ = 509203 has got this property and also all multipliers k_n = k₀ + 11184810 · n (n ≥ 1).
Such numbers are called Riesel numbers. The Riesel problem is finding the smallest Riesel number.

Conjecture: k = 509203 is the smallest Riesel number.

To prove this conjecture, for every k < 509203 a prime k · 2ⁿ-1 must be found. It's sufficient to find the first exponent n giving a prime.
So the search is separated in stages f_m where the first exponent giving a prime is in the interval 2^m ≤ n < 2^m+1.

Data :

49 remaining k's with no prime for n < 8.95M

k-values
2293, 9221, 23669, 31859, 38473, 46663, 67117, 74699, 81041, 93839, 97139, 107347, 121889, 129007, 143047, 146561, 161669, 192971, 206039, 206231, 215443, 226153, 234343, 245561, 250027, 315929, 319511, 324011, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, 368411, 371893, 384539, 386801, 397027, 409753, 444637, 470173, 474491, 477583, 485557, 494743

Number of k's prime for first prime in the interval 2^m ≤ n < 2^m+1. (green: all these k's are listed in the summary pages)

m	2^m	2^m+1	# k's	contributors(primes found)	Date completed (last prime found)
0	1	2	39867	W.Keller	1992
1	2	4	59460	W.Keller	1992
2	4	8	62311	W.Keller	1992
3	8	16	45177	W.Keller	1992
4	16	32	24478	W.Keller	1992
5	32	64	11668	W.Keller	1992
6	64	128	5360	W.Keller	1992
7	128	256	2728	W.Keller	1992
8	256	512	1337	W.Keller	1992
9	512	1024	785	W.Keller	1992
10	1024	2048	467	W.Keller	1992
11	2048	4096	289	Y.Gallot	1998
12	4096	8192	191	Y.Gallot	1998
13	8192	16384	125	Y.Gallot	1998-05-11
14	16384	32768	87	R.Ballinger(21), W.Keller(25), S.Key(41)	1998-11-14
15	32768	65536	62	R.Ballinger(5), K.J.Brazier(1), C.Caldwell(2), W.Keller(13), S.Key(22), T.Kuechler(1), D.W.Linton(18)	1999-04-09
16	65536	131072	38	R.Ballinger(2), Gallot(1), W.Keller(4), S.Key(5), D.W.Linton(25), H.Zeisel(1)	1999-10-22
17	131072	262144	35	A.Aitsen(1), R.Ballinger(2), L.Baxter(1), Y.Gallot(1), O.Haeberle(1), R.A.A.Heylen(1), D.W.Linton(25), P.Pirson(1), J.Szmidt(1), H.Zeisel(1)	2001-07-13
18	262144	524288	25	D.Andrews(1), R.Ballinger(4), K.Davis(1), O.Haeberle(1), R.A.A.Heylen(2), R.Keiser(1), T.Kuechler(1), N.Kuosa(1), D.W.Linton(6), P.Pirson(1), M.Rodenkirch(1), L.Schmid(1), J.Szmidt(2), J.Wolfe(1), H.Zeisel(1)	2003-12-04
19	524288	1048576	22	O.Haeberle(10), R.Ballinger(1), R.A.A.Heylen(1), D.W.Linton(1), L.Schmid(1), RieselSieve(7), D.Domanov(1)	2012-02-02
20	1048576	2097152	18	D.W.Linton(1), R.Ballinger(1), RieselSieve(16)	2007-10-28
21	2097152	4194304	13	RieselSieve(8), PrimeGrid(5)	2011-05-08
22	4194304	8388608	8	PrimeGrid(8)	2014-10-04
23	>8388608	Infinity	1 (49 remain)	PrimeGrid(1) (see PrimeGrid at n = 8.95M)	2017-12-13

f(13) = 125 where k·2ⁿ-1 prime for 8192 ≤ n < 16384

k	n	k	n	k	n	k	n	k	n	k	n	k	n	k	n
8681	8458	13451	10562	16033	8547	19507	9713	24959	8976	29819	13052	36761	12018	43543	9419
44669	8560	46439	15088	46901	9346	50227	11893	50729	14788	50801	8970	52777	8333	58753	11955
59473	10187	64213	13839	65519	11148	76237	13085	85933	14383	86437	8933	94439	12424	94589	13436
96367	9409	97723	9447	108817	9429	115151	10930	122879	8812	126389	15576	132217	15281	136421	13534
139183	8419	156101	10798	157957	16065	164921	13258	165491	14486	166483	16243	168907	9225	175121	13402
183523	15387	186059	14932	192581	11006	194527	11573	194819	15256	195875	9350	204397	8673	205781	8334
207311	8998	209647	8561	212321	13838	213553	12771	217499	11244	220829	15960	220931	8930	225689	10800
228509	9980	236893	14751	237659	15856	239887	13473	241987	11277	245563	10319	246731	12078	247597	9917
248047	11613	250091	15690	253507	10833	260261	16202	263669	9864	266459	8852	282395	11218	283481	9394
288943	9227	290707	8457	291463	10791	293647	11225	295303	11711	299909	8828	300089	15160	301267	12493
303689	10916	305377	11457	306637	10273	307631	8254	310843	10775	312869	16116	317969	14300	322877	8328
325571	10610	326047	8801	335353	14591	345967	10269	348037	9113	352123	14995	353375	9382	355693	10203
363119	9712	363359	9788	364079	9052	369163	9099	373573	9167	375097	12421	379693	9655	400093	12815
401969	14732	414373	10703	415559	11444	417937	14725	420239	9488	424631	13166	430439	15600	434099	10420
435269	16060	437009	10196	449143	13347	457213	12839	459701	8706	463961	9794	471547	15141	482711	15046
489367	11393	491017	9141	501209	10008	502979	9044	507613	15019

f(14) = 87 where k·2ⁿ-1 prime for 16384 ≤ n < 32768

k	n	k	n	k	n	k	n	k	n	k	n	k	n	k	n
1211	25242	10391	31914	10451	19246	14347	25997	22889	16692	24839	20436	25825	26961	34631	25390
63463	17219	63877	21409	69059	31344	69301	32437	79273	21107	85919	28516	90595	21643	95411	28454
97919	20872	114223	22463	115915	16389	125707	19021	129119	19584	131227	30441	131321	16910	142619	18716
148151	25790	149467	22949	164981	17806	169709	18856	171607	21117	177073	26147	180133	19879	184031	23906
190979	23244	194563	24343	207839	29140	218651	30314	227441	24590	228071	27662	230017	20045	233227	16517
236377	19693	244841	17890	262391	19766	278713	17159	281891	29950	300611	21194	302047	19737	306749	18916
313777	16713	315433	23183	318463	17707	320503	26627	323579	30324	323737	18001	328687	25357	329413	16947
343079	22092	347201	27574	350173	25523	350617	31493	353999	16932	359923	22463	360817	18285	367543	17831
368467	29221	370991	18522	372311	28974	376303	18335	384791	22598	387223	20951	391679	17432	397679	27540
403537	27505	406423	18155	410803	31895	419729	19632	435101	20382	439811	20290	447367	22149	450193	23059
456959	20016	462967	24757	483907	18089	484241	22158	486677	27910	505649	18644	508979	17088

f(15) = 62 where k·2ⁿ-1 prime for 32768 ≤ n < 65536

k	n	k	n	k	n	k	n	k	n	k	n	k	n	k	n
3817	40381	16829	57728	25079	34660	33569	52512	44107	65173	57311	40498	57359	62500	64387	40249
81313	64619	82987	49489	86833	44747	104549	39744	107069	43280	111883	54415	116531	59782	133211	34342
137951	33094	140921	33234	152729	48348	164789	48268	175379	37676	177479	41032	209227	43481	213719	54996
220511	54890	233833	32983	254587	42953	264403	35147	278077	56913	288907	60485	289409	37444	297581	62982
300389	34768	300637	60109	308351	45326	314203	38267	318431	40618	320531	46318	338321	49538	341443	42575
348059	34660	353557	51833	358243	40707	381193	43627	382229	39004	385627	34229	401879	39864	404527	37265
406253	40948	410489	47596	423403	34691	424499	34168	431173	63643	438071	55986	453433	49725	453581	58266
461861	61762	465089	33868	468311	59978	478837	55081	487811	42478	497647	59825

f(16) = 38 where k·2ⁿ-1 prime for 65536 ≤ n < 131072

k	n	k	n	k	n	k	n	k	n	k	n	k	n
30689	78560	41983	72347	48703	109415	77099	69356	83323	83079	95561	76754	103811	114034
124439	94176	124663	68939	131623	69099	138097	86401	150103	111391	162943	74319	170069	96828
179743	99731	182939	127180	217963	95111	224891	92214	245533	84667	275377	89937	296321	101594
301151	78326	302111	96282	332201	103498	358889	122900	362707	119925	364331	92626	392317	74281
392923	70827	399281	67798	404429	85344	420067	125733	444847	112497	462079	105187	464909	73740
467639	74496	477311	105214	483479	115376

f(17) = 35 where k·2ⁿ-1 prime for 131072 ≤ n < 262144

k	n	k	n	k	n	k	n	k	n	k	n	k	n
11519	164444	14459	171144	25229	238652	37837	180873	81517	258321	105569	235200	105697	223233
107167	159161	111253	165379	111763	155551	130297	136645	132071	202098	132599	206032	144817	258857
159821	168770	178747	144789	185767	149009	190229	141576	217807	243537	256267	148941	281143	187639
285191	201138	307211	241978	321043	238303	325859	156148	331139	201240	370421	201442	392737	248517
393209	221216	408247	205469	438523	135415	466783	245839	471127	157629	485773	216487	499031	139894

f(18) = 25 where k·2ⁿ-1 prime for 262144 ≤ n < 524288

f(19) = 22 where k·2ⁿ-1 prime for 524288 ≤ n < 1048576

f(20) = 18 where k·2ⁿ-1 prime for 1048576 < n < 2097152

primes
71009·2^1185112-1	110413·2^1591999-1	149797·2^1414137-1	150847·2^1076441-1	152713·2^1154707-1	192089·2^1395688-1
234847·2^1535589-1	325627·2^1472117-1	345067·2^1876573-1	350107·2^1144101-1	357659·2^1779748-1	412717·2^1084409-1
417643·2^1800787-1	467917·2^1993429-1	469949·2^1649228-1	500621·2^1138518-1	502541·2^1199930-1	504613·2^1136459-1

f(21) = 13 where k·2ⁿ-1 prime for 2097152 ≤ n < 4194304

primes
26773·2^2465343-1	65531·2^3629342-1	113983·2^3201175-1	114487·2^2198389-1	123547·2^3804809-1	191249·2^3417696-1
196597·2^2178109-1	275293·2^2335007-1	342673·2^2639439-1	415267·2^3771929-1	428639·2^3506452-1	450457·2^2307905-1
485767·2^3609357-1

f(22) = 8 where k·2ⁿ-1 prime for 4194304 ≤ n < 8388608

primes
141941·2^4299438-1	252191·2^5497878-1	353159·2^4331116-1	398023·2^6418059-1	304207·2^6643565-1	40597·2^6808509-1
402539·2^7173024-1	502573·2^7181987-1

f(23) ≥ 1 where k·2ⁿ-1 prime for 8388608 ≤ n < 16777216

primes
273809·2^8932416-1

primes
27253·2²⁷²³⁴⁷-1	39269·2²⁸⁷⁰⁴⁸-1	42779·2³²²⁹⁰⁸-1	43541·2⁵⁰⁷⁰⁹⁸-1	46271·2⁴²⁸²¹⁰-1	104917·2³⁴⁰¹⁸¹-1
130139·2²⁸⁰²⁹⁶-1	144643·2⁴⁹⁸⁰⁷⁹-1	148901·2³⁶⁰³³⁸-1	159371·2²⁸⁴¹⁶⁶-1	189463·2³²⁴¹⁰³-1	201193·2⁴⁵⁷⁶¹⁵-1
220063·2³⁰⁶³³⁵-1	235601·2²⁹⁵³³⁸-1	245051·2²⁸⁵⁷⁵⁰-1	267763·2²⁶⁴¹¹⁵-1	277153·2⁴²⁹⁸¹⁹-1	299617·2⁴²⁸⁹¹⁷-1
376993·2²⁹³⁶⁰³-1	382691·2⁴³¹⁷²²-1	398533·2⁴¹⁹¹⁰⁷-1	401617·2⁴⁷⁰¹⁴⁹-1	416413·2⁴²⁴⁷⁹¹-1	443857·2³⁶⁹⁴⁵⁷-1
465869·2⁴⁹⁷⁵⁹⁶-1

primes
659·2⁸⁰⁰⁵¹⁶-1	89707·2⁵⁷⁸³¹³-1	93997·2⁸⁶⁴⁴⁰¹-1	98939·2⁵⁷⁵¹⁴⁴-1	103259·2⁶¹⁵⁰⁷⁶-1	109897·2⁶³⁰²²¹-1
126667·2⁶²⁶⁴⁹⁷-1	162941·2⁹⁹³⁷¹⁸-1	170591·2⁸⁶⁶⁸⁷⁰-1	204223·2⁶⁹⁶⁸⁹¹-1	212893·2⁷³⁰³⁸⁷-1	215503·2⁶⁴⁹⁸⁹¹-1
220033·2⁷¹⁹⁷³¹-1	222997·2⁶¹³¹⁵³-1	246299·2⁷⁵²⁶⁰⁰-1	261221·2⁶⁸⁹⁴²²-1	279703·2⁶¹⁶²³⁵-1	309817·2⁹⁰¹¹⁷³-1
357491·2⁶⁰⁹³³⁸-1	401143·2⁵³²⁹²⁷-1	458743·2⁵⁴⁷⁷⁹¹-1	460139·2⁷⁷⁹⁵³⁶-1

The Riesel-Problem

Definition :

Data :

49 remaining k's with no prime for n < 8.95M

f(13) = 125 where k·2n-1 prime for 8192 ≤ n < 16384

f(14) = 87 where k·2n-1 prime for 16384 ≤ n < 32768

f(15) = 62 where k·2n-1 prime for 32768 ≤ n < 65536

f(16) = 38 where k·2n-1 prime for 65536 ≤ n < 131072

f(17) = 35 where k·2n-1 prime for 131072 ≤ n < 262144

f(18) = 25 where k·2n-1 prime for 262144 ≤ n < 524288

f(19) = 22 where k·2n-1 prime for 524288 ≤ n < 1048576

f(20) = 18 where k·2n-1 prime for 1048576 < n < 2097152

f(21) = 13 where k·2n-1 prime for 2097152 ≤ n < 4194304

f(22) = 8 where k·2n-1 prime for 4194304 ≤ n < 8388608

f(23) ≥ 1 where k·2n-1 prime for 8388608 ≤ n < 16777216