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. allksrieselbase2.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. allksrieselbase22ndconj.zip 

2 evenn 
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 LiskovetsGallot conjectures. allksrieselbase2evenn.txt 
2 oddn 
172677  5, 7, 13, 17, 241  All k where k = 2*m^2: let k = 2*m^2 and let n = 2*q1; 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 LiskovetsGallot conjectures. allksrieselbase2oddn.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. allksrieselbase4.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) 
allksrieselbase8.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. allksrieselbase16.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) 
allksrieselbase32.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. allksrieselbase64.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) 
allksrieselbase128.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. allksrieselbase256.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) 
allksrieselbase512.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. allksrieselbase1024.txt 