Woodall primes and Near Woodall Primes

Started: 2007-07-18 Last update: 2014-07-30

Latest News:

2014-07-29: new Near Woodall prime was found: 454483·2^1817935-1 = 1817932·2^1817933-1.
2012-11-14: new Near Woodall prime was found: 582833·2^1165668-1 = 1165666·2^1165667-1.
2012-10-03: new Near Woodall prime was found: 586085·2^1172172-1 = 1172170·2^1172171-1.
2010-11-17: new Near Woodall prime was found: 197673·2⁷⁹⁰⁶⁹⁵-1 = 790692·2⁷⁹⁰⁶⁹³-1.
2010-02-09: new Near Woodall prime was found: 665127·2⁶⁶⁵¹²⁸-1.
2010-02-01: new Near Woodall prime was found: 41197·2⁶⁵⁹¹⁵⁷-1 = 659152·2⁶⁵⁹¹⁵³-1.
2009-09-11: new Near Woodall prime was found!
2008-09-25: new Near Woodall prime was found!

See reserved and checked ranges for Woodall Search here.
See the Top 5000 Top 20 Woodall Primes here.
A list of Near Woodall Primes / Near Cullen Primes can be found here.

Woodall Primes

A Woodall prime is any prime of the form n·2ⁿ-1. It is conjectured that there are infinitely many Woodall primes.

History:
1968: H.Riesel listet all Woodall Primes for n≤110
1984: W.Keller searched for n≤15000 (some Woodall Primes found by others but not listed as)
1987: W.Keller searched for n<20000

Note: The Woodall Primes for n=2 and n=512 are Mersenne Primes!

There are 33 Woodall primes discovered so far :

This table contains all Woodall Primes with n<5000000.

no.	n	#digits	normalized form	who found	date
1	2	1	2³-1	-	-
2	3	2	3·2³-1	-	-
3	6	3	3·2⁷-1	-	-
4	30	11	15·2³¹-1	H.Riesel	1968
5	75	25	75·2⁷⁵-1	H.Riesel	1968
6	81	27	81·2⁸¹-1	H.Riesel	1968
7	115	37	115·2¹¹⁵-1	H.C.Williams, C.R.Zarnke	1968
8	123	40	123·2¹²³-1	W.Keller	1984
9	249	78	249·2²⁴⁹-1	W.Keller	1984
10	362	112	181·2³⁶³-1	I.Jönsson	1972
11	384	119	3·2³⁹¹-1	H.Riesel	1969
12	462	142	231·2⁴⁶³-1	W.Keller	1984
13	512	157	2⁵²¹-1	R.M.Robinson	1952-01-30
14	751	229	751·2⁷⁵¹-1	W.Keller	1984
15	822	251	411·2⁸²³-1	W.Keller	1984
16	5312	1603	83·2⁵³¹⁸-1	W.Keller	1984-12
17	7755	2339	7755·2⁷⁷⁵⁵-1	W.Keller	1984-12
18	9531	2874	9531·2⁹⁵³¹-1	W.Keller	1984-12
19	12379	3731	12379·2¹²³⁷⁹-1	W.Keller	1984-12
20	15822	4768	7911·2¹⁵⁸²³-1	W.Keller	1987-12
21	18885	5690	18885·2¹⁸⁸⁸⁵-1	W.Keller	1987-12
22	22971	6920	22971·2²²⁹⁷¹-1	J.Young	1997-05
23	23005	6930	23005·2²³⁰⁰⁵-1	J.Young	1997-05
24	98726	29725	49363·2⁹⁸⁷²⁷-1	J.Young	1997-05
25	143018	43058	71509·2¹⁴³⁰¹⁹-1	R.Ballinger	1998-04-25
26	151023	45468	151023·2¹⁵¹⁰²³-1	K.O'Hare	1998-05-01
27	667071	200815	667071·2⁶⁶⁷⁰⁷¹-1	M.Toplic	2000-09-25
28	1195203	359799	1195203·2^1195203-1	M.Rodenkirch	2005-07-11
29	1268979	382007	1268979·2^1268979-1	W.Siemelink	2007-01-25
30	1467763	441847	1467763·2^1467763-1	W.Siemelink, Z.Mate, M.Rodenkirch	2007-06-07
31	2013992	606279	251749·2^2013995-1	L.M.Andersen	2007-08-09
32	2367906	712818	1183953·2^2367907-1	S.Kohlman	2007-09-03
33	3752948	1129757	938237·2^3752950-1	M.J.Thompson	2007-12-26

Near Woodall primes (form (n+1)·2ⁿ-1):

Note: The Near Woodall Primes for n=1, n=3 and n=15 are Mersenne Primes!

This table contains all Near Woodall Primes with n<1100000.

no.	n	#digits	normalized form	who found	date
1	1	1	2²-1	n.n.	n.n.
2	2	2	3·2²-1	n.n.	n.n.
3	3	2	2⁵-1	n.n.	n.n.
4	4	2	5·2⁴-1	n.n.	n.n.
5	5	3	3·2⁶-1	n.n.	n.n.
6	9	4	5·2¹⁰-1	n.n.	n.n.
7	14	6	15·2¹⁴-1	n.n.	n.n.
8	15	6	2¹⁹-1	n.n.	n.n.
9	16	7	17·2¹⁶-1	n.n.	n.n.
10	27	10	7·2²⁹-1	n.n.	n.n.
11	45	16	23·2⁴⁶-1	n.n.	n.n.
12	122	39	123·2¹²²-1	n.n.	n.n.
13	125	40	63·2¹²⁶-1	n.n.	n.n.
14	213	67	107·2²¹⁴-1	n.n.	n.n.
15	242	76	243·2²⁴²-1	n.n.	n.n.
16	256	80	257·2²⁵⁶-1	n.n.	n.n.
17	263	82	33·2²⁶⁶-1	n.n.	n.n.
18	290	90	291·2²⁹⁰-1	n.n.	n.n.
19	855	261	107·2⁸⁵⁸-1	n.n.	n.n.
20	1059	322	265·2¹⁰⁶¹-1	n.n.	n.n.
21	2273	688	1137·2²²⁷⁴-1	n.n.	n.n.
22	3945	1192	1973·2³⁹⁴⁶-1	n.n.	n.n.
23	3999	1208	125·2⁴⁰⁰⁴-1	n.n.	n.n.
24	9512	2868	9513·2⁹⁵¹²-1	H.Lifchitz	1998-07-03
25	14127	4257	883·2¹⁴¹³¹-1	H.Lifchitz	1998-07-03
26	16486	4967	16487·2¹⁶⁴⁸⁶-1	H.Lifchitz	1998-07-03
27	20056	6042	20057·2²⁰⁰⁵⁶-1	H.Lifchitz	1998-07-03
28	28834	8685	28835·2²⁸⁸³⁴-1	H.Lifchitz	1998-07-03
29	41493	12496	20747·2⁴¹⁴⁹⁴-1	H.Lifchitz	1998-07-03
30	159147	47914	39787·2¹⁵⁹¹⁴⁹-1	R.Ballinger	2000-05-19
31	227139	68382	56785·2²²⁷¹⁴¹-1	R.Ballinger	2000-07-28
32	587823	176959	36739·2⁵⁸⁷⁸²⁷-1	S.Harvey	2009-09-11

Near Woodall primes (form (n-1)·2ⁿ-1):

Note: The Near Woodall Primes for n=2 and n=5 are Mersenne Primes!

This table contains all Near Woodall Primes with n<1100000.

no.	n	#digits	normalized form	who found	date
1	2	1	2²-1	n.n.	n.n.
2	4	2	3·2⁴-1	n.n.	n.n.
3	5	3	2⁷-1	n.n.	n.n.
4	11	5	5·2¹²-1	n.n.	n.n.
5	28	10	27·2²⁸-1	n.n.	n.n.
6	35	13	17·2³⁶-1	n.n.	n.n.
7	235	74	117·2²³⁶-1	n.n.	n.n.
8	325	101	81·2³²⁷-1	n.n.	n.n.
9	551	169	275·2⁵⁵²-1	n.n.	n.n.
10	688	210	687·2⁶⁸⁸-1	n.n.	n.n.
11	7037	2123	1759·2⁷⁰³⁹-1	n.n.	n.n.
12	8896	2682	8895·2⁸⁸⁹⁶-1	n.n.	n.n.
13	10020	3021	10019·2¹⁰⁰²⁰-1	M.Oakes	n.n.
14	13597	4098	3399·2¹³⁵⁹⁹-1	M.Oakes	n.n.
15	21347	6431	10673·2²¹³⁴⁸-1	H.Lifchitz	1998-07-03
16	118020	35533	118019·2¹¹⁸⁰²⁰-1	S.Harvey	2002-06-27
17	446694	134474	446693·2⁴⁴⁶⁶⁹⁴-1	S.Harvey	2008-09-25
18	659153	198431	41197·2⁶⁵⁹¹⁵⁷-1	S.Harvey	2010-02-01
19	665128	200230	665127·2⁶⁶⁵¹²⁸-1	S.Harvey	2010-02-09
20	790693	238029	197673·2⁷⁹⁰⁶⁹⁵-1	S.Harvey	2010-11-17
21	1165667	350907	582833·2^1165668-1	S.Harvey	2012-11-14
22	1172171	352865	586085·2^1172172-1	S.Harvey	2012-10-03
23	1817933	547259	454483·2^1817935-1	S.Harvey	2014-07-29

Woodall primes and Near Woodall Primes

Woodall Primes

Near Woodall primes (form (n+1)·2n-1):

Near Woodall primes (form (n-1)·2n-1):

Near Woodall primes (form (n+1)·2ⁿ-1):

Near Woodall primes (form (n-1)·2ⁿ-1):