The Liskovets-Gallot Conjectures

All data taken from the forum-threads
Riesel/Sierp base 2 even-k/even-n/odd-n testing upto post #352 (2016-08-16),
R/S base 2 even-k/even-n/odd-n - PRPnet mini-drive upto post #154 (2011-08-10) and
PRPnet Servers for CRUS.

Definition :

Valery Liskovets studied the list of k·2ⁿ+1 primes and observed, that the k's (k divisible by 3)
got an irregular contribution of odd and even exponents yielding a prime.

For example for k=51 there are 37 odd n values and 7 even for k·2ⁿ+1 and
k=87 got 37 even and 2 odd n values for k·2ⁿ+1.
Yves Gallot extended this for k·2ⁿ-1 numbers.

References:
Problems & Puzzles: Problem 36. The Liskovets-Gallot numbers.
Definition of conjectures opinions needed...
Sierpinski base 4 Forum.

Conjectures:
The following odd k's (divisible by 3) are the smallest such k where k·2ⁿ+1 / k·2ⁿ-1
can be prime only for odd n / even n values:

Yves Gallot asserted first the following numbers are the smallest without proof:

Even Sierpinski k·2ⁿ+1 composite for all even n: k=66741 down here (Status: 39 primes found, all done)
Odd Sierpinski k·2ⁿ+1 composite for all odd n: k=95283 down here (Status: 56 primes found, 3 to go)
Even Riesel k·2ⁿ-1 composite for all even n: k=39939 down here (Status: 21 primes found, 2 to go)
Odd Riesel k·2ⁿ-1 composite for all odd n: k=172677 down here (Status: 98 primes found, 4 to go)

This page reflects the progress of proving these 4 conjectures.

Note:
To prove that a special odd k yields a prime for one even n = 2m it's adequate to test k·4^m-1/+1,
and to prove that a special odd k yields a prime for one odd n = 2m+1, it's adequate to test 2k·4^m-1/+1.

Countdown: 9

Overview of most recent found primes and number of primes left to prove the four conjectures.

left	date	contributor	prime found / to do
			Riesel even k=9519
			Riesel even k=14361
			Riesel odd k=39687
			Riesel odd k=103947
			Riesel odd k=154317
			Riesel odd k=163503
			Sierpinski odd k=9267
			Sierpinski odd k=32247
			Sierpinski odd k=53133
9	2015-08-03	J.Penné	23451·2^3739388+1
10	2014-06-13	G.Barnes	19401·2^3086450-1
11	2014-11-08	G.Barnes	155877·2^2273465-1
12	2014-10-20	G.Barnes	84363·2^2222321+1
13	2014-06-09	J.Penné	60849·2^3067914+1
14	2011-07-29	M.Dettweiler	148323·2^1973319-1
15	2011-07-08	L.Vogel	85287·2^1890011+1
16	2011-06-02	I.M.Gunn	20049·2^1687252-1
17	2011-06-01	M.Dettweiler	60357·2^1676907+1
18	2009-01-19	K.Bonath	147687·2⁸⁴³⁶⁸⁹-1
19	2008-12-05	J.Penné	133977·2⁸¹¹⁴⁸⁵-1
20	2008-12-03	K.Bonath	30003·2⁶¹³⁴⁶³-1
21	2008-11-17	K.Bonath	106377·2⁴⁷⁵⁵⁶⁹-1
22	2008-05-23	G.Barnes	6927·2⁷⁴³⁴⁸¹-1
23	2008-04-07	J.Penné	145257·2⁴⁴³⁰⁷⁷-1
24	2008-04-02	K.Bonath	172167·2²⁸²⁶⁴⁹-1
25	2008-03-31	G.Barnes	80463·2⁴⁶⁸¹⁴¹+1
26	2008-03-31	K.Bonath	99363·2²⁶⁸⁸⁷⁹-1
27	2008-03-31	K.Bonath	130467·2²⁷³⁴³⁷-1
28	2008-03-28	J.Penné	86613·2³⁵⁶⁹⁶⁷-1
29	2008-02-18	J.Penné	144117·2²²⁴⁹⁷⁷-1
30	2008-02-14	G.Barnes	37953·2²⁹⁸⁹¹³+1
31	2008-02-11	G.Barnes	70467·2²⁶⁸⁵⁰³+1

Riesel candidates (k·2ⁿ-1, even n):

Testing k·4ⁿ-1 (alternatively k·2ⁿ-1 only even n)
2 remaining k's and 21 primes found

k	contributor	last edit	prime / [range tested]
2181	J.Penné	2008-01-19	37890
6549	J.Penné	2008-01-19	5076
8181	J.Penné	2008-01-19	8018
8961	J.Penné	2008-01-19	30950
9519	(PuzzlePeter)	2017-10-18	[16.0M]
11379	J.Penné	2008-01-19	32252
12849	J.Penné	2008-01-19	9788
14361	CRUS	2017-10-18	[5.14M]
14859	J.Penné	2008-01-19	11228
15639	J.Penné	2008-01-19	66328
16431	J.Penné	2008-01-19	4198
17889	J.Penné	2008-01-19	10628
19401	G.Barnes	2015-06-13	3086450
20049	I.M.Gunn	2011-06-02	1687252
21501	J.Penné	2008-01-19	7286
26091	J.Penné	2008-01-19	4198
26511	J.Penné	2008-01-19	167154
26601	J.Penné	2008-01-19	46246
30171	J.Penné	2008-01-19	76286
31431	J.Penné	2008-01-19	16942
31749	J.Penné	2008-01-19	5040
31959	J.Penné	2008-01-19	19704
35259	J.Penné	2008-01-19	10540
39939	Riesel number without even prime n

Riesel candidates (k·2ⁿ-1, odd n):

4 remaining k's and 98 primes found

k	contributor	last edit	prime / [range tested]
903	J.Penné	2008-01-20	10227
4887	J.Penné	2008-01-20	4289
5007	J.Penné	2008-01-20	6765
5163	J.Penné	2008-01-20	6183
6927	G.Barnes	2008-05-23	743481
7977	J.Penné	2008-01-20	31265
8367	G.Barnes	2008-01-28	313705
9087	J.Penné	2008-01-20	4741
10113	J.Penné	2008-01-20	14535
15213	J.Penné	2008-01-20	20311
19377	J.Penné	2008-01-20	18677
21813	J.Penné	2008-01-20	4283
22863	J.Penné	2008-01-20	101135
27957	J.Penné	2008-01-20	21477
30003	K.Bonath	2008-12-03	613463
30357	J.Penné	2008-01-20	65361
32937	J.Penné	2008-01-20	8473
33837	J.Penné	2008-01-20	4273
34533	J.Penné	2008-01-20	32899
35193	J.Penné	2008-01-20	12483
39687	CRUS	2017-10-18	[5.14M]
44283	J.Penné	2008-01-20	4439
46107	J.Penné	2008-01-20	4277
46923	J.Penné	2008-01-22	65175
48927	J.Penné	2008-01-21	35861
52137	J.Penné	2008-01-20	26309
53973	K.Bonath	2008-01-21	198575
55983	J.Penné	2008-01-20	9851
56493	J.Penné	2008-01-20	6891
59655	J.Penné	2008-01-21	43825
59763	J.Penné	2008-01-20	4611
61833	J.Penné	2008-01-20	4651
63153	J.Penné	2008-01-20	60295
64023	J.Penné	2008-01-20	11431
67737	J.Penné	2008-01-20	4437
70743	J.Penné	2008-01-20	49387
72327	J.Penné	2008-01-20	17125
72993	J.Penné	2008-01-20	23319
75093	J.Penné	2008-01-20	15371
75363	J.Penné	2008-01-22	120595
75387	J.Penné	2008-01-20	5181
75873	J.Penné	2008-01-22	62419
78933	J.Penné	2008-01-20	11443
79437	J.Penné	2008-01-22	35093
84807	J.Penné	2008-01-20	47389
86613	J.Penné	2008-03-28	356967
87735	J.Penné	2008-01-20	4551
88623	J.Penné	2008-01-20	13251
88743	J.Penné	2008-01-20	4619
90567	J.Penné	2008-01-20	6577
91671	J.Penné	2008-01-20	8795
93507	J.Penné	2008-01-20	5449
97323	J.Penné	2008-01-20	52207
99363	K.Bonath	2008-03-31	268879
100053	J.Penné	2008-01-20	28459
100353	J.Penné	2008-01-20	5147
100377	J.Penné	2008-01-28	231813
101823	J.Penné	2008-01-20	4519
102993	J.Penné	2008-01-20	48975
103947	CRUS	2017-10-18	[5.14M]
105123	J.Penné	2008-01-20	5555
105837	J.Penné	2008-01-20	5913
106377	K.Bonath	2008-11-17	475569
114249	J.Penné	2008-01-22	48469
115167	J.Penné	2008-01-20	8685
117303	J.Penné	2008-01-20	4451
117867	J.Penné	2008-01-20	4513
120387	J.Penné	2008-01-20	5645
121557	J.Penné	2008-01-20	11817
129747	J.Penné	2008-01-20	18657
130383	J.Penné	2008-01-24	104123
130467	K.Bonath	2008-03-31	273437
131727	J.Penné	2008-01-24	169621
132507	J.Penné	2008-01-20	4485
133023	J.Penné	2008-01-20	9087
133977	J.Penné	2008-12-05	811485
134037	J.Penné	2008-01-20	4421
135567	J.Penné	2008-01-24	68325
142683	J.Penné	2008-01-20	22371
144117	J.Penné	2008-02-18	224977
144393	J.Penné	2008-01-20	6567
144957	J.Penné	2008-01-20	6473
145257	J.Penné	2008-04-07	443077
147687	K.Bonath	2009-01-19	843689
148227	J.Penné	2008-01-20	5997
148323	M.Dettweiler	2011-07-29	1973319
148803	J.Penné	2008-01-20	25019
152907	J.Penné	2008-01-20	4365
154317	CRUS	2017-10-18	[5.14M]
154827	J.Penné	2008-01-20	9113
155877	G.Barnes	2014-11-10	2273465
157383	J.Penné	2008-01-20	44059
161583	J.Penné	2008-01-28	138710
163503	CRUS	2017-10-18	[5.14M]
167007	J.Penné	2008-01-20	4901
167997	J.Penné	2008-01-20	18705
169527	J.Penné	2008-01-20	9329
169743	J.Penné	2008-01-20	23791
170223	J.Penné	2008-01-20	4187
170733	J.Penné	2008-01-20	7307
171783	J.Penné	2008-01-20	6759
172167	K.Bonath	2008-04-02	282649
172677	Riesel number without odd prime n

Sierpinski candidates (k·2ⁿ+1, even n):

Testing k·4ⁿ+1 (alternatively k·2ⁿ+1 only even n)
0 remaining k's and 39 primes found

k	contributor	last edit	prime / [range tested]
2379			8114
8139	H.Aggarwal	2005-02-01	25954
9609			5422
10281			7444
11709			6882
12711			5092
14661	J.Penné	2005-01-27	91368
15441	D.Wallace	2005-01-27	20584
17169			6450
21069	D.Wallace	2005-01-27	23006
21699	D.Wallace	2005-01-27	72874
23451	J.Penné	2015-08-03	3739388
23799	K.J.Brazier	2005-01-29	105890
23901			11292
30579	(Mark)	2005-01-28	48594
33771	J.Penné	2005-01-29	178200
33879	D.Wallace	2005-06-20	378022
35889			7770
39231			13716
39759			4594
40269			8458
41289			13514
41709	G.Reynolds	2005-01-29	80594
42717	J.Penné	2005-10-14	905792
44469			13134
51171	T.Masser	2005-01-31	93736
52419			4578
52701			6976
52839	T.Masser	2005-01-27	32558
53979			7590
55611	H.Aggarwal	2005-01-29	40212
56019			8094
56139			4858
58791	(Mystwalker)	2005-01-27	79420
60849	J.Penné	2014-02-14	3067914
60891	J.Penné	2005-01-29	40144
61371			12576
62391			5472
63411	D.Wallace	2005-01-31	72064
66741	Sierpinski number without even prime n

Sierpinski candidates (k·2ⁿ+1, odd n):

3 remaining k's and 56 primes found

k	contributor	last edit	prime / [range tested]
93	J.Penné	2008-01-15	20917
2943	J.Penné	2005-02-06	108041
5193	J.Penné	2008-01-15	4277
5703	J.Penné	2008-01-15	5149
5823	J.Penné	2008-01-15	8105
6807	J.Penné	2008-01-15	4415
7233	J.Penné	2008-01-15	4277
9267	J.Penné	2017-10-18	[8406703]
9777	J.Penné	2008-01-15	18975
10923	J.Penné	2008-01-15	6801
14397	J.Penné	2008-01-15	4347
16917	J.Penné	2008-01-15	12799
17457	T.Cadigan	2005-01-28	29563
17937	T.Cadigan	2005-01-28	53927
20997	J.Penné	2008-01-15	8191
24693	D.Wallace	2005-05-25	357417 from Sierpinski base 4
25083	(Mark)	2005-01-27	24981 from Sierpinski base 4
25917	J.Penné	2008-01-15	9671
26613	J.Penné	2008-01-15	89749
30933	J.Penné	2008-01-15	4433
32247	CRUS	2017-10-18	[5.14M]
35787	J.Penné	2008-01-15	36639
37953	G.Barnes	2008-02-14	298913
38463	G.Barnes	2008-01-15	58753
39297	G.Barnes	2008-01-20	169495
40857	J.Penné	2008-01-15	5383
42993	J.Penné	2008-01-15	16165
43167	J.Penné	2008-01-15	9795
46623	J.Penné	2008-01-15	79553
49563	J.Penné	2008-01-15	5813
50433	G.Barnes	2008-01-20	156597
53133	CRUS	2017-10-18	[5.14M]
60273	J.Penné	2008-01-15	7421
60357	M.Dettweiler	2011-06-01	1676907
60963	G.Barnes	2008-01-15	73409
61137	G.Barnes	2008-01-20	162967
62307	G.Barnes	2008-01-15	44559
63357	J.Penné	2008-01-15	4211
65253	J.Penné	2008-01-15	10301
67917	J.Penné	2008-01-15	13079
69963	J.Penné	2008-01-15	5205
70467	G.Barnes	2008-02-11	268503
72537	J.Penné	2008-01-15	15771
73023	J.Penné	2008-01-15	17965
75183	G.Barnes	2008-01-15	35481
78543	J.Penné	2008-01-15	10089
78753	G.Barnes	2008-01-15	63761
80463	G.Barnes	2008-03-31	468141
80517	J.Penné	2008-01-15	5423
81147	J.Penné	2008-01-15	17615
82197	J.Penné	2008-01-15	5079
84363	G.Barnes	2014-10-20	2222321
85287	L.Vogel	2011-06-02	1890011
88863	J.Penné	2008-01-15	9825
91383	J.Penné	2008-01-15	15333
91437	G.Barnes	2008-01-20	161615
93033	J.Penné	2008-01-15	30473
93477	G.Barnes	2008-01-15	63251
93663	G.Barnes	2008-01-15	82317
95283	Sierpinski number without odd prime n

The Liskovets-Gallot Conjectures

Definition :

Countdown: 9

Riesel candidates (k·2n-1, even n):

Riesel candidates (k·2n-1, odd n):

Sierpinski candidates (k·2n+1, even n):

Sierpinski candidates (k·2n+1, odd n):

Riesel candidates (k·2ⁿ-1, even n):

Riesel candidates (k·2ⁿ-1, odd n):

Sierpinski candidates (k·2ⁿ+1, even n):

Sierpinski candidates (k·2ⁿ+1, odd n):