Sierpinski conjectures and proofs
Powers of 2

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

Compiled by Gary Barnes

Riesel conjectures
Riesel conjectures powers of 2
Riesel conjecture reservations
Sierpinski conjectures
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 4 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.  Make generalized Fermat numbers (GFn's), i.e. q^m*b^n+1 where m>=0 and q is a root of the base.
4.  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 Sierpinski 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 / GFn's without a prime / accounting of all k's
2 78557 3, 5, 7, 13, 19, 37, 73   none 21181 (41.4M)
22699 (41.8M)
24737 (41.5M)
55459 (41.6M)
67607 (41.4M)
10223 (31172165)
19249 (13018586)
27653 (9167433)
28433 (7830457)
33661 (7031232)
5359 (5054502)
4847 (3321063)
54767 (1337287)
69109 (1157446)
65567 (1013803)
All k's are being worked on by PrimeGrid's Seventeen or Bust project.  See k's and test limits at Seventeen or Bust stats.

k = 65536 is a GFn with no known prime.

all-ks-sierp-base2.txt
2
2nd conjecture
271129 3, 5, 7, 13, 17, 241   none 79309 (31.9M)
79817 (31.9M)
90646 (10M)
91549 (26.6M)
101746 (10M)
131179 (26.6M)
152267 (32M)
156511 (32.1M)
163187 (26.1M)
200749 (26.6M)
209611 (26.7M)
222113 (31.9M)
225931 (31.7M)
227723 (26.1M)
229673 (26.1M)
237019 (31.5M)
238411 (26.6M)
202705 (21320516)
168451 (19375200)
99739 (14019102)
193997 (11452891)
90527 (9162167)
161041 (7107964)
258317 (5450519)
265711 (4858008)
211195 (3224974)
94373 (3206717)
Only 78557<k<271129 are considered.

Prime k's are being worked on by PrimeGrid's Prime Sierpinski Problem project.  See k's and test limits at Prime Sierpinski stats.

Composite odd k's are being worked on by PrimeGrid's Extended Sierpinski Problem project.  See k's and test limits at Extended Sierpinski stats.

k = 131072 and 262144 are GFn's with no known prime.

all-ks-sierp-base2-2nd-conj.zip
2
even-n
66741 5, 7, 13, 17, 241   k = = 2 mod 3 (3) none - proven 23451 (3739388)
60849 (3067914)
42717 (905792)
33879 (378022)
33771 (178200)
23799 (105890)
51171 (93736)
14661 (91368)
41709 (80594)
58791 (79420)
Only k's where k = = 3 mod 6 are considered.

See additional details at The Liskovets-Gallot conjectures.

all-ks-sierp-base2-evenn.txt
2
odd-n
95283 5, 7, 13, 19, 73, 109   k = = 1 mod 3 (3) 9267 (15.7M)
32247 (10M)
53133 (10M)
84363 (2222321)
85287 (1890011)
60357 (1676907)
80463 (468141)
24693 (357417)
37953 (298913)
70467 (268503)
39297 (169495)
61137 (162967)
91437 (161615)
Only k's where k = = 3 mod 6 are considered.

See additional details at The Liskovets-Gallot conjectures.

all-ks-sierp-base2-oddn.txt
4 66741 5, 7, 13, 17, 241   k = = 2 mod 3 (3) 18534 (7.85M)
21181 (20.7M)
22699 (20.9M)
49474 (20.75M)
55459 (20.8M)
64494 (5M)
20446 (15586082)
19249 (6509293)
55306 (4583716)
56866 (3915228)
33661 (3515616)
5359 (2527251)
23451 (1869694)
9694 (1660531)
60849 (1533957)
44131 (497986)
k's where k = = 1 mod 3 are being worked on by PrimeGrid's Seventeen or Bust project.  k's, test limits, and primes are converted from base 2.

k = 65536 is a GFn with no known prime.

all-ks-sierp-base4.txt
8 47 3, 5, 13 All k = m^3 for all n; factors to:
(m*2^n + 1) *
(m^2*4^n - m*2^n + 1)
k = = 6 mod 7 (7) none - proven 31 (20)
46 (4)
40 (4)
37 (4)
28 (4)
45 (3)
38 (3)
36 (3)
26 (3)
23 (3)
k = 1 and 8 proven composite by full algebraic factors.

all-ks-sierp-base8.txt
16 66741 7, 13, 17, 241 All k = 4*q^4 for all n:
   let k = 4*q^4
   and let m=q*2^n; factors to:
     (2*m^2 + 2m + 1) *
     (2*m^2 - 2m + 1)
k = = 2 mod 3 (3)
k = = 4 mod 5 (5)
2908 (1M)
6663 (1M)
10183 (1M)
17118 (1M)
21181 (10.35M)
24582 (1M)
30397 (1M)
35818 (1M)
40410 (1M)
42745 (1M)
44035 (1M)
57867 (1M)
60070 (1M)
64620 (1M)
20446 (7793041)
55306 (2291858)
56866 (1957614)
33661 (1757808)
21436 (1263625)
23451 (934847)
65077 (901486)
38776 (830265)
53653 (577962)
18598 (484327)
k's where k = = 1 mod 15 are being worked on by PrimeGrid's Seventeen or Bust project.  k's, test limits, and primes are converted from base 2.

k = 2500 and 40000 proven composite by full algebraic factors.

k = 65536 is a GFn with no known prime.

all-ks-sierp-base16.txt
32 10 3, 11 All k = m^5 for all n; factors to:
(m*2^n + 1) *
(m^4*16^n - m^3*8^n + m^2*4^n - m*2^n + 1)
k = = 30 mod 31 (31) none - proven 9 (13)
7 (4)
5 (3)
6 (1)
3 (1)
k = 1 proven composite by full algebraic factors.

k = 4 is a GFn with no known prime.

all-ks-sierp-base32.txt
64 51 5, 13 All k = m^3 for all n; factors to:
(m*4^n + 1) *
(m^2*16^n - m*4^n + 1)
k = = 2 mod 3 (3)
k = = 6 mod 7 (7)
none - proven 24 (31)
31 (10)
30 (6)
39 (3)
46 (2)
45 (2)
40 (2)
37 (2)
36 (2)
28 (2)
k = 1 proven composite by full algebraic factors.

all-ks-sierp-base64.txt
128 44 3, 43 All k = m^7 for all n; factors to:
(m*2^n + 1) *
(m^6*64^n - m^5*32^n + m^4*16^n - m^3*8^n + m^2*4^n - m*2^n + 1)
k = = 126 mod 127 (127) 40 (1.2857M) 41 (39271)
42 (13001)
20 (473)
28 (322)
38 (291)
19 (178)
25 (64)
3 (27)
17 (21)
31 (20)
k = 1 proven composite by full algebraic factors.

k = 16 is a GFn with no known prime.

k = 8 and 32 are GFn's with no possible prime.

all-ks-sierp-base128.txt
256 1221 7, 13, 241   k = = 2 mod 3 (3)
k = = 4 mod 5 (5)
k = = 16 mod 17 (17)
831 (1M) 535 (109243)
691 (25890)
712 (19406)
946 (6821)
346 (2914)
1165 (2368)
751 (1914)
1132 (1763)
523 (1428)
888 (1360)
all-ks-sierp-base256.txt
512 18 5, 13, 19 All k = m^3 for all n; factors to:
(m*8^n + 1) *
(m^2*64^n - m*8^n + 1)
k = = 6 mod 7 (7)
k = = 72 mod 73 (73)
5 (1M) 12 (23)
14 (21)
7 (20)
11 (9)
9 (7)
10 (6)
17 (3)
3 (2)
15 (1)
k = 1 and 8 proven composite by full algebraic factors.

k = 2, 4, and 16 are GFn's with no known prime.

all-ks-sierp-base512.txt
1024 81 5, 41 All k = m^5 for all n; factors to:
(m*4^n + 1) *
(m^4*256^n - m^3*64^n + m^2*16^n - m*4^n + 1)
k = = 2 mod 3 (3)
k = = 10 mod 11 (11)
k = = 30 mod 31 (31)
none - proven 9 (323)
51 (266)
33 (142)
48 (53)
24 (35)
55 (22)
52 (8)
45 (6)
31 (6)
34 (5)
k = 1 proven composite by full algebraic factors.

k = 4 and 16 are GFn's with no known prime.

all-ks-sierp-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