# 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)

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

## Countdown: 9

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

leftdatecontributorprime 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
92015-08-03J.Penné23451·23739388+1
102014-06-13G.Barnes19401·23086450-1
112014-11-08G.Barnes155877·22273465-1
122014-10-20G.Barnes84363·22222321+1
132014-06-09J.Penné60849·23067914+1
142011-07-29M.Dettweiler148323·21973319-1
152011-07-08L.Vogel85287·21890011+1
162011-06-02I.M.Gunn20049·21687252-1
172011-06-01M.Dettweiler60357·21676907+1
182009-01-19K.Bonath147687·2843689-1
192008-12-05J.Penné133977·2811485-1
202008-12-03K.Bonath30003·2613463-1
212008-11-17K.Bonath106377·2475569-1
222008-05-23G.Barnes6927·2743481-1
232008-04-07J.Penné145257·2443077-1
242008-04-02K.Bonath172167·2282649-1
252008-03-31G.Barnes80463·2468141+1
262008-03-31K.Bonath99363·2268879-1
272008-03-31K.Bonath130467·2273437-1
282008-03-28J.Penné86613·2356967-1
292008-02-18J.Penné144117·2224977-1
302008-02-14G.Barnes37953·2298913+1
312008-02-11G.Barnes70467·2268503+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
kcontributorlast editprime / [range tested]
2181J.Penné2008-01-1937890
6549J.Penné2008-01-195076
8181J.Penné2008-01-198018
8961J.Penné2008-01-1930950
9519(PuzzlePeter)2017-10-18[16.0M]
11379J.Penné2008-01-1932252
12849J.Penné2008-01-199788
14361CRUS2017-10-18[5.14M]
14859J.Penné2008-01-1911228
15639J.Penné2008-01-1966328
16431J.Penné2008-01-194198
17889J.Penné2008-01-1910628
19401G.Barnes2015-06-133086450
20049I.M.Gunn2011-06-021687252
21501J.Penné2008-01-197286
26091J.Penné2008-01-194198
26511J.Penné2008-01-19167154
26601J.Penné2008-01-1946246
30171J.Penné2008-01-1976286
31431J.Penné2008-01-1916942
31749J.Penné2008-01-195040
31959J.Penné2008-01-1919704
35259J.Penné2008-01-1910540
39939Riesel number without even prime n

## Riesel candidates (k·2n-1, odd n):

4 remaining k's and 98 primes found
kcontributorlast editprime / [range tested]
903J.Penné2008-01-2010227
4887J.Penné2008-01-204289
5007J.Penné2008-01-206765
5163J.Penné2008-01-206183
6927G.Barnes2008-05-23743481
7977J.Penné2008-01-2031265
8367G.Barnes2008-01-28313705
9087J.Penné2008-01-204741
10113J.Penné2008-01-2014535
15213J.Penné2008-01-2020311
19377J.Penné2008-01-2018677
21813J.Penné2008-01-204283
22863J.Penné2008-01-20101135
27957J.Penné2008-01-2021477
30003K.Bonath2008-12-03613463
30357J.Penné2008-01-2065361
32937J.Penné2008-01-208473
33837J.Penné2008-01-204273
34533J.Penné2008-01-2032899
35193J.Penné2008-01-2012483
39687CRUS2017-10-18[5.14M]
44283J.Penné2008-01-204439
46107J.Penné2008-01-204277
46923J.Penné2008-01-2265175
48927J.Penné2008-01-2135861
52137J.Penné2008-01-2026309
53973K.Bonath2008-01-21198575
55983J.Penné2008-01-209851
56493J.Penné2008-01-206891
59655J.Penné2008-01-2143825
59763J.Penné2008-01-204611
61833J.Penné2008-01-204651
63153J.Penné2008-01-2060295
64023J.Penné2008-01-2011431
67737J.Penné2008-01-204437
70743J.Penné2008-01-2049387
72327J.Penné2008-01-2017125
72993J.Penné2008-01-2023319
75093J.Penné2008-01-2015371
75363J.Penné2008-01-22120595
75387J.Penné2008-01-205181
75873J.Penné2008-01-2262419
78933J.Penné2008-01-2011443
79437J.Penné2008-01-2235093
84807J.Penné2008-01-2047389
86613J.Penné2008-03-28356967
87735J.Penné2008-01-204551
88623J.Penné2008-01-2013251
88743J.Penné2008-01-204619
90567J.Penné2008-01-206577
91671J.Penné2008-01-208795
93507J.Penné2008-01-205449
97323J.Penné2008-01-2052207
99363K.Bonath2008-03-31268879
100053J.Penné2008-01-2028459
100353J.Penné2008-01-205147
100377J.Penné2008-01-28231813
101823J.Penné2008-01-204519
102993J.Penné2008-01-2048975
103947CRUS2017-10-18[5.14M]
105123J.Penné2008-01-205555
105837J.Penné2008-01-205913
106377K.Bonath2008-11-17475569
114249J.Penné2008-01-2248469
115167J.Penné2008-01-208685
117303J.Penné2008-01-204451
117867J.Penné2008-01-204513
120387J.Penné2008-01-205645
121557J.Penné2008-01-2011817
129747J.Penné2008-01-2018657
130383J.Penné2008-01-24104123
130467K.Bonath2008-03-31273437
131727J.Penné2008-01-24169621
132507J.Penné2008-01-204485
133023J.Penné2008-01-209087
133977J.Penné2008-12-05811485
134037J.Penné2008-01-204421
135567J.Penné2008-01-2468325
142683J.Penné2008-01-2022371
144117J.Penné2008-02-18224977
144393J.Penné2008-01-206567
144957J.Penné2008-01-206473
145257J.Penné2008-04-07443077
147687K.Bonath2009-01-19843689
148227J.Penné2008-01-205997
148323M.Dettweiler2011-07-291973319
148803J.Penné2008-01-2025019
152907J.Penné2008-01-204365
154317CRUS2017-10-18[5.14M]
154827J.Penné2008-01-209113
155877G.Barnes2014-11-102273465
157383J.Penné2008-01-2044059
161583J.Penné2008-01-28138710
163503CRUS2017-10-18[5.14M]
167007J.Penné2008-01-204901
167997J.Penné2008-01-2018705
169527J.Penné2008-01-209329
169743J.Penné2008-01-2023791
170223J.Penné2008-01-204187
170733J.Penné2008-01-207307
171783J.Penné2008-01-206759
172167K.Bonath2008-04-02282649
172677Riesel 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
kcontributorlast editprime / [range tested]
2379  8114
8139H.Aggarwal2005-02-0125954
9609  5422
10281  7444
11709  6882
12711  5092
14661J.Penné2005-01-2791368
15441D.Wallace2005-01-2720584
17169  6450
21069D.Wallace2005-01-2723006
21699D.Wallace2005-01-2772874
23451J.Penné2015-08-033739388
23799K.J.Brazier2005-01-29105890
23901  11292
30579(Mark)2005-01-2848594
33771J.Penné2005-01-29178200
33879D.Wallace2005-06-20378022
35889  7770
39231  13716
39759  4594
40269  8458
41289  13514
41709G.Reynolds2005-01-2980594
42717J.Penné2005-10-14905792
44469  13134
51171T.Masser2005-01-3193736
52419  4578
52701  6976
52839T.Masser2005-01-2732558
53979  7590
55611H.Aggarwal2005-01-2940212
56019  8094
56139  4858
58791(Mystwalker)2005-01-2779420
60849J.Penné2014-02-143067914
60891J.Penné2005-01-2940144
61371  12576
62391  5472
63411D.Wallace2005-01-3172064
66741Sierpinski number without even prime n

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

3 remaining k's and 56 primes found
kcontributorlast editprime / [range tested]
93J.Penné2008-01-1520917
2943J.Penné2005-02-06108041
5193J.Penné2008-01-154277
5703J.Penné2008-01-155149
5823J.Penné2008-01-158105
6807J.Penné2008-01-154415
7233J.Penné2008-01-154277
9267J.Penné2017-10-18[8406703]
9777J.Penné2008-01-1518975
10923J.Penné2008-01-156801
14397J.Penné2008-01-154347
16917J.Penné2008-01-1512799
20997J.Penné2008-01-158191
24693D.Wallace2005-05-25357417 from Sierpinski base 4
25083(Mark)2005-01-2724981 from Sierpinski base 4
25917J.Penné2008-01-159671
26613J.Penné2008-01-1589749
30933J.Penné2008-01-154433
32247CRUS2017-10-18[5.14M]
35787J.Penné2008-01-1536639
37953G.Barnes2008-02-14298913
38463G.Barnes2008-01-1558753
39297G.Barnes2008-01-20169495
40857J.Penné2008-01-155383
42993J.Penné2008-01-1516165
43167J.Penné2008-01-159795
46623J.Penné2008-01-1579553
49563J.Penné2008-01-155813
50433G.Barnes2008-01-20156597
53133CRUS2017-10-18[5.14M]
60273J.Penné2008-01-157421
60357M.Dettweiler2011-06-011676907
60963G.Barnes2008-01-1573409
61137G.Barnes2008-01-20162967
62307G.Barnes2008-01-1544559
63357J.Penné2008-01-154211
65253J.Penné2008-01-1510301
67917J.Penné2008-01-1513079
69963J.Penné2008-01-155205
70467G.Barnes2008-02-11268503
72537J.Penné2008-01-1515771
73023J.Penné2008-01-1517965
75183G.Barnes2008-01-1535481
78543J.Penné2008-01-1510089
78753G.Barnes2008-01-1563761
80463G.Barnes2008-03-31468141
80517J.Penné2008-01-155423
81147J.Penné2008-01-1517615
82197J.Penné2008-01-155079
84363G.Barnes2014-10-202222321
85287L.Vogel2011-06-021890011
88863J.Penné2008-01-159825
91383J.Penné2008-01-1515333
91437G.Barnes2008-01-20161615
93033J.Penné2008-01-1530473
93477G.Barnes2008-01-1563251
93663G.Barnes2008-01-1582317
95283Sierpinski number without odd prime n