Riesel conjectures & proofs powers of 2 (original) (raw)
Started: Dec. 21, 2007
Last update: Oct. 13, 2025
Requirements and inclusions:
1. Conjectures must have a finite covering set and cannot be a MOB.
2. Primes must be n >= 1.
3. k-values that are a MOB are included in the conjectures but some may be excluded. See exclusion 3 below.
k-values with any of the following conditions are excluded from the conjectures:
1. All n-values are covered by one trivial factor.
2. All n-values are covered by algebraic factors or a combination of algebraic and trivial factor(s).
3. k is a MOB and k-1 is not prime. They will have the same prime as k / b.
Green = testing through other projects
Gray = conjecture proven
Testing not done through other projects is coordinated atMersenneforum Conjectures 'R Us.
Base
Conjectured Riesel k
Covering set
k's that make a full covering set with all or partial algebraic factors
Trivial k's (factor)
Remaining k to find prime
(n testing limit)
Top 10 k's with largest first primes: k (n)
Comments / accounting of all k's
2
509203
3, 5, 7, 13, 17, 241
none
43 k's remaining at n>=10M.
See k's and test limits at Riesel Base 2 remain.
107347 (23427517)
97139 (18397548)
93839 (15337656)
192971 (14773498)
206039 (13104952)
2293 (12918431)
9221 (11392194)
146561 (11280802)
273809 (8932416)
502573 (7181987)
All odd k's are being worked on by PrimeGrid'sRiesel Problem project. See k's and test limits at Riesel Problem stats.
2
2nd conjecture
762701
3, 5, 7, 13, 17, 241
none
527587 (8.8M)
539069 (8.8M)
548441 (8.8M)
578593 (8.8M)
581053 (8.8M)
587759 (8.8M)
589783 (8.8M)
593219 (8.8M)
600461 (8.8M)
606674 (8.8M)
611773 (8.8M)
613211 (8.8M)
641579 (8.8M)
671413 (8.8M)
672071 (8.8M)
681041 (8.8M)
685183 (8.8M)
686711 (8.8M)
704501 (8.8M)
705329 (8.8M)
746321 (8.8M)
747133 (8.8M)
755857 (8.8M)
542714 (8943012)
580633 (7208783)
625783 (7031319)
554051 (6517658)
521921 (6101122)
519397 (4908893)
612749 (4254500)
543131 (3529754)
700057 (3113753)
582971 (3053414)
Only 509203<k<762701 are considered.
all-ks-riesel-base2-2nd-conj.zip
2
even-n
39939
5, 7, 13, 19, 73, 109
All k where k = m^2:
let k = m^2
and let n = 2*q; factors to:
(m*2^q - 1) *
(m*2^q + 1)
k = = 1 mod 3 (3)
9519 (16.777M)
14361 (10M)
19401 (3086450)
20049 (1687252)
26511 (167154)
30171 (76286)
15639 (66328)
26601 (46246)
2181 (37890)
11379 (32252)
8961 (30950)
31959 (19704)
Only k's where k = = 3 mod 6 are considered.
k = 3^2, 9^2, 15^2, (etc. repeating every 6m) proven composite by full algebraic factors.
See additional details atThe Liskovets-Gallot conjectures.
2
odd-n
172677
5, 7, 13, 17, 241
All k where k = 2*m^2:
let k = 2*m^2
and let n = 2*q-1; factors to:
(m*2^q - 1) *
(m*2^q + 1)
k = = 2 mod 3 (3)
39687 (10M)
103947 (10M)
154317 (10M)
163503 (10M)
155877 (2273465)
148323 (1973319)
147687 (843689)
133977 (811485)
6927 (743481)
30003 (613463)
106377 (475569)
145257 (443077)
86613 (356967)
8367 (313705)
Only k's where k = = 3 mod 6 are considered.
No k's proven composite by algebraic factors.
See additional details atThe Liskovets-Gallot conjectures.
4
39939
5, 7, 13, 19, 73, 109
All k = m^2 for all n; factors to:
(m*2^n - 1) *
(m*2^n + 1)
k = = 1 mod 3 (3)
9519 (8.388M)
14361 (5M)
19464 (5M)
23669 (8.275M)
31859 (8.275M)
4586 (6459215)
9221 (5696097)
19401 (1543225)
20049 (843626)
659 (400258)
13854 (371740)
16734 (156852)
39269 (143524)
25229 (119326)
14459 (85572)
k = 3^2, 6^2, 9^2, (etc. repeating every 3m) proven composite by full algebraic factors.
k's where k = = 2 mod 3 are being worked on by PrimeGrid'sRiesel Problem project. k's, test limits, and primes are converted from base 2.
8
14
3, 5, 13
k = = 1 mod 7 (7)
none - proven
11 (18)
5 (4)
12 (3)
7 (3)
2 (2)
13 (1)
10 (1)
9 (1)
6 (1)
4 (1)
16
33965
7, 13, 17, 241
All k = m^2 for all n; factors to:
(m*4^n - 1) *
(m*4^n + 1)
k = = 1 mod 3 (3)
k = = 1 mod 5 (5)
443 (2M)
2297 (2M)
9519 (4.194M)
13380 (2M)
13703 (2M)
19464 (2.5M)
19772 (2M)
23669 (4.1375M)
24987 (2M)
26378 (2M)
28967 (2M)
29885 (2M)
31859 (4.1375M)
33023 (2M)
18344 (3229607)
21555 (1841032)
12587 (615631)
3620 (435506)
20049 (421813)
7673 (366247)
33863 (236436)
6852 (216571)
2993 (211161)
15068 (204680)
k = 3^2, 12^2, 15^2, 18^2, 27^2, 30^2, (etc. pattern repeating every 30m) proven composite by full algebraic factors.
k's where k = = 14 mod 15 are being worked on by PrimeGrid'sRiesel Problem project. k's, test limits, and primes are converted from base 2.
32
10
3, 11
k = = 1 mod 31 (31)
none - proven
3 (11)
2 (6)
9 (3)
8 (2)
5 (2)
7 (1)
6 (1)
4 (1)
64
14
5, 13
All k = m^2 for all n; factors to:
(m*8^n - 1) *
(m*8^n + 1)
k = = 1 mod 3 (3)
k = = 1 mod 7 (7)
none - proven
11 (9)
12 (6)
5 (2)
6 (1)
3 (1)
2 (1)
k = 9 proven composite by full algebraic factors.
128
44
3, 43
k = = 1 mod 127 (127)
none - proven
29 (211192)
23 (2118)
26 (1442)
37 (699)
16 (459)
42 (246)
35 (98)
30 (66)
36 (59)
12 (46)
256
10364
7, 13, 241
All k = m^2 for all n; factors to:
(m*16^n - 1) *
(m*16^n + 1)
k = = 1 mod 3 (3)
k = = 1 mod 5 (5)
k = = 1 mod 17 (17)
659 (1M)
807 (1M)
1695 (1M)
1808 (1M)
2237 (1M)
2297 (1M)
2759 (1M)
4377 (1M)
4559 (1M)
5768 (1M)
7088 (1M)
7130 (1M)
7673 (1M)
7968 (1M)
8087 (1M)
8334 (1M)
8765 (1M)
9519 (2.097M)
10154 (1M)
6332 (748660)
5502 (400821)
5903 (367343)
7335 (336135)
7913 (284458)
10110 (280347)
7890 (236791)
3480 (231670)
3620 (217753)
6213 (206892)
k = 9, 144, 225, 729, 900, 1764, 2025, 2304, 3249, 3600, 3969, 5184, 5625, 6084, 7569, 8100, and 8649 proven composite by full algebraic factors.
512
14
3, 5, 13
k = = 1 mod 7 (7)
k = = 1 mod 73 (73)
none - proven
4 (2215)
13 (2119)
9 (7)
11 (6)
6 (6)
5 (2)
3 (2)
2 (2)
12 (1)
10 (1)
1024
81
5, 41
All k = m^2 for all n; factors to:
(m*32^n - 1) *
(m*32^n + 1)
k = = 1 mod 3 (3)
k = = 1 mod 11 (11)
k = = 1 mod 31 (31)
29 (1M)
74 (666084)
39 (4070)
65 (93)
69 (54)
3 (47)
71 (41)
44 (36)
26 (29)
68 (25)
59 (16)
k = 9 and 36 proven composite by full algebraic factors.