Riesel conjectures and proofs
Powers of 2

Started: Dec. 21, 2007
Last update: May 28, 2024

Compiled by Gary Barnes

Riesel conjectures
Riesel conjecture reservations
Sierpinski conjectures
Sierpinski conjectures powers of 2
Sierpinski conjecture reservations

Abbreviations:  b = base, MOB = multiple of the base

Conjecture requirements:
1.  Conjectures must have a finite covering set.
2.  2nd and subsequent conjectures cannot have a factor that is a power of the base.

General requirements and inclusions:
1.  Primes must be n >= 1.
2.  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 at Mersenneforum 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 44 k's remaining at n>=10M.

See k's and test limits at Riesel Base 2 remain.
97139 (18397548)
93839 (15337656)
192971 (14773498)
206039 (13104952)
2293 (12918431)
9221 (11392194)
146561 (11280802)
273809 (8932416)
502573 (7181987)
402539 (7173024)
All odd k's are being worked on by PrimeGrid's Riesel Problem project.  See k's and test limits at Riesel Problem stats.

all-ks-riesel-base2.zip
2
2nd conjecture
762701 3, 5, 7, 13, 17, 241   none 527587 (8M)
539069 (8M)
542714 (8M)
548441 (8M)
578593 (8M)
581053 (8M)
587759 (8M)
589783 (8M)
593219 (8M)
600461 (8M)
606674 (8M)
611773 (8M)
613211 (8M)
641579 (8M)
671413 (8M)
672071 (8M)
681041 (8M)
685183 (8M)
686711 (8M)
704501 (8M)
705329 (8M)
746321 (8M)
747133 (8M)
755857 (8M)
580633 (7208783)
625783 (7031319)
554051 (6517658)
521921 (6101122)
519397 (4908893)
612749 (4254500)
543131 (3529754)
700057 (3113753)
582971 (3053414)
543539 (2536028)
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 at The Liskovets-Gallot conjectures.

all-ks-riesel-base2-evenn.txt
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 at The Liskovets-Gallot conjectures.

all-ks-riesel-base2-oddn.txt
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 (7.9M)
31859 (7.9M)
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's Riesel Problem project.  k's, test limits, and primes are converted from base 2.

all-ks-riesel-base4.txt
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)
all-ks-riesel-base8.txt
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 (1.5M)
2297 (1.5M)
9519 (4.194M)
13380 (1M)
13703 (1M)
19464 (2.5M)
19772 (1M)
21555 (1M)
23669 (3.95M)
24987 (1M)
26378 (1M)
28967 (1M)
29885 (1M)
31859 (3.95M)
33023 (1M)
18344 (3229607)
12587 (615631)
3620 (435506)
20049 (421813)
7673 (366247)
33863 (236436)
6852 (216571)
2993 (211161)
15068 (204680)
659 (200129)
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's Riesel Problem project.  k's, test limits, and primes are converted from base 2.

all-ks-riesel-base16.txt
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)
all-ks-riesel-base32.txt
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.

all-ks-riesel-base64.txt
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)
all-ks-riesel-base128.txt
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 (750K)
807 (750K)
1695 (750K)
1808 (750K)
2237 (750K)
2297 (750K)
2759 (750K)
4377 (750K)
4559 (750K)
5768 (750K)
7088 (750K)
7130 (750K)
7673 (750K)
7968 (750K)
8087 (750K)
8334 (750K)
8765 (750K)
9519 (2.097M)
10154 (750K)
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.

all-ks-riesel-base256.txt
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)
all-ks-riesel-base512.txt
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.

all-ks-riesel-base1024.txt


Original information obtained from:
Mersenneforum Prime Search Projects Conjectures 'R Us threads:
   Sierpinski/Riesel-Base 22
   Sierpinski/Riesel bases 6 to 18
   Sierpinski base 4
   Sierpinski/Riesel Base 10
   Sierpinski/Riesel-Base 23
   Even k's and the Riesel conjecture
   Even k's and the Sierpinski conjecture
Mersenneforum Prime Search Projects Sierpinski/Riesel Base 5 forum
Riesel Prime database
Riesel Problem project
Seventeen or Bust project
Top 5000 primes

Contributors:
CRUS contributors