Sierpinski conjectures and proofs
Powers of 2

Started: Dec. 21, 2007
Last update: Nov. 3, 2018

Compiled by Gary Barnes

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

All n must be >= 1.

k-values with at least one of the following conditions are excluded from the conjectures:
     1.  All n-values have a single trivial factor.
     2.  Make a full covering set with all or partial algebraic factors.
     3.  Make generalized Fermat numbers (GFn's), i.e. q^m*b^n+1 where b is the base, m>=0, and q is a root of the base.

k-values that are a multiple of base (b) and where k+1 is composite are included in the conjectures but excluded from testing.
Such k-values 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 and GFn's without a prime
2 78557 3, 5, 7, 13, 19, 37, 73   none 21181 (31.6M)
22699 (31.6M)
24737 (31.6M)
55459 (31.6M)
67607 (31.6M)
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.
2
2nd conjecture
271129 3, 5, 7, 13, 17, 241   78557 (3, 5, 7, 13, 19, 37, 73) 21181 (31.6M)
22699 (31.6M)
24737 (31.6M)
55459 (31.6M)
67607 (31.6M)
79309 (20M)
79817 (20M)
90646 (5.5M)
91549 (12.3M)
99739 (12.3M)
101746 (5.5M)
131179 (12.3M)
152267 (20M)
156511 (20M)
163187 (12.3M)
200749 (12.3M)
202705 (12.3M)
209611 (12.3M)
222113 (20M)
225931 (20M)
227723 (12.3M)
229673 (12.3M)
237019 (20M)
238411 (12.3M)
10223 (31172165)
168451 (19375200)
19249 (13018586)
193997 (11452891)
27653 (9167433)
90527 (9162167)
28433 (7830457)
161041 (7107964)
33661 (7031232)
258317 (5450519)
k<78557 is being worked on by PrimeGrid's Seventeen or Bust project.  See k's and test limits at Seventeen or Bust stats.

Prime k>78557 is being worked on by PrimeGrid's Prime Sierpinski Problem project.  See k's and test limits at Prime Sierpinski stats.

Composite odd k>78557 is being worked on by PrimeGrid's Extended Sierpinski Problem project.  See k's and test limits at Extended Sierpinski stats.

k = 65536, 131072, and 262144 are GFn's with no known prime.
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.
2
odd-n
95283 5, 7, 13, 19, 73, 109   k = = 1 mod 3 (3) 9267 (10.1M)
32247 (5.5M)
53133 (5.5M)
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.
4 66741 5, 7, 13, 17, 241   k = = 2 mod 3 (3) 18534 (5.07M)
21181 (15.8M)
22699 (15.8M)
49474 (15.8M)
55459 (15.8M)
64494 (2.75M)
20446 (15586082)
19249 (6509293)
55306 (4583716)
56866 (3915228)
33661 (3515616)
5359 (2527251)
23451 (1869694)
9694 (1660531)
60849 (1533957)
44131 (497986)
k's where k = = 0 mod 3 were originally the Sierpinski Base 4 project but are now coordinated through CRUS.

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.
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.
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 (750K)
6663 (750K)
10183 (750K)
17118 (750K)
21181 (7.9M)
24582 (750K)
30397 (750K)
35818 (750K)
40410 (750K)
42745 (750K)
44035 (750K)
57867 (750K)
60070 (750K)
64620 (750K)
65077 (750K)
20446 (7793041)
55306 (2291858)
56866 (1957614)
33661 (1757808)
21436 (1263625)
23451 (934847)
38776 (830265)
53653 (577962)
18598 (484327)
31347 (411467)
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's where k = = 6 mod 15 were originally the Sierpinski Base 4 project but are now coordinated through CRUS.  k's, test limits, and primes are converted from base 4.

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

k=65536 is a GFn with no known prime.
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.
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.
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.
256 1221 7, 13, 241   k = = 2 mod 3 (3)
k = = 4 mod 5 (5)
k = = 16 mod 17 (17)
831 (800K) 535 (109243)
691 (25890)
712 (19406)
946 (6821)
346 (2914)
1165 (2368)
751 (1914)
1132 (1763)
523 (1428)
888 (1360)
 
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.
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.

 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


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