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 (38.3M) 22699 (38.2M) 24737 (38.2M) 55459 (38.1M) 67607 (38.1M) 
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  none  79309 (28.2M) 79817 (27.8M) 90646 (10M) 91549 (23M) 101746 (10M) 131179 (23M) 152267 (28.2M) 156511 (27.8M) 163187 (23M) 200749 (23M) 209611 (23M) 222113 (28.2M) 225931 (27.9M) 227723 (23M) 229673 (23M) 237019 (27.9M) 238411 (23M) 
202705 (21320516) 168451 (19375200) 99739 (14019102) 193997 (11452891) 90527 (9162167) 161041 (7107964) 258317 (5450519) 265711 (4858008) 211195 (3224974) 94373 (3206717) 
Only k>78557 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. 

2 evenn 
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 LiskovetsGallot conjectures. 

2 oddn 
95283  5, 7, 13, 19, 73, 109  k = = 1 mod 3 (3)  9267 (14.7M) 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 LiskovetsGallot conjectures. 

4  66741  5, 7, 13, 17, 241  k = = 2 mod 3 (3)  18534 (7.35M) 21181 (19.15M) 22699 (19.1M) 49474 (19.1M) 55459 (19.05M) 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. 

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 (1M) 6663 (1M) 10183 (1M) 17118 (1M) 21181 (9.575M) 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. 
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 (1M)  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. 