A Sierpinski/Riesel-like problem

[sweety439]

R792 is also an interesting base, as there are many k's with algebraic factorization (combine of difference of squares and one-covering)

CK = 365, covering set = {13, 61}

tested to n=2000

(Condition 1):
All k where k = m^2
and m = = 5 or 8 mod 13:
   for even n let k = m^2
   and let n = 2*q; factors to:
     (m*792^q - 1) *
     (m*792^q + 1)
   odd n:
     factor of 13
(Condition 2):
All k where k = m^2
and m = = 11 or 50 mod 61:
   for even n let k = m^2
   and let n = 2*q; factors to:
     (m*792^q - 1) *
     (m*792^q + 1)
   odd n:
     factor of 61
(Condition 3):
All k where k = 22*m^2
and m = = 4 or 9 mod 13:
   even n:
     factor of 13
   for odd n let k = 22*m^2
   and let n=2*q-1; factors to:
     [m*6^n*22^q - 1] *
     [m*6^n*22^q + 1]
(Condition 4):
All k where k = 22*m^2
and m = = 5 or 56 mod 61:
   even n:
     factor of 61
   for odd n let k = 22*m^2
   and let n=2*q-1; factors to:
     [m*6^n*22^q - 1] *
     [m*6^n*22^q + 1]

k = 25, 64, 324 proven composite by condition 1.

k = 121 proven composite by condition 2.

k = 352 proven composite by condition 3.

[sweety439]

Originally Posted by sweety439

now I start R515 and R536

R515 has CK = 5 (for all bases b == 14 mod 15, 4 is both Sierpinski and Riesel, and for all bases b == 11 mod 12, 5 is both Sierpinski and Riesel)

thus this base is trivial, only k = 1 through k = 4 need primes, however, only k = 3 has an easy prime (n<1000).

1,2243
2,58466
3,68
4,1579

Note that R515 has no single prime with n <= 60 (I doubt whether it is the smallest such base ... but this needs to check)

[sweety439]

Take the following bases:

R165
R178 (k=19)
R186

[sweety439]

Originally Posted by sweety439

Take the following bases:

R165
R178 (k=19)
R186

All tested to n=6K, no prime or PRP found

bases released, reserve R181 (k=21)

[sweety439]

Originally Posted by sweety439

Take the following bases:

R165
R178 (k=19)
R186

R165 at n=15K
R186 at n=13K

[sweety439]

R178 update

k=19 has PRP at n=13655
k=4 remains at n=13K, continuing ....

update for R85, k=61 at n=11K

[sweety439]

status update:

R85 at n=15K
R31 k=5 at n=30K

[sweety439]

R181 k=21 at n=12K

reserve R181 k=5

[sweety439]

R181 k=5 at n=21000, no prime or PRP

Reserve S103 (k=7)

[sweety439]

S103 k=7 at n=22000, no primes or PRPs

[sweety439]

All 1k bases (I have a plan to reserve all of them to 20K+ or 30K+)

base   remain k   test limit of n
S12   12   33.55M
S18   18   33.55M
S25   71   10K
S32   4   1.717G
S37   37   524K
S38   1   16.77M
S50   1   16.77M
S53   4   2M
S55   1   524K
S62   1   16.77M
S72   72   16.77M
S77   1   524K
S89   1   524K
S91   1   524K
S92   1   16.77M
S98   1   16.77M
S99   1   524K
S103   7   22K
S104   1   16.77M
S107   1   524K
S109   1   524K
S113   17   8K
S118   48   740K
S140   8   1M
S143   1   524K
S144   1   16.77M
S149   1   524K
S151   1   524K
S160   20   2K
S165   43   2K
S174   4   1M
S176   55   2K
S179   1   524K
S189   1   524K
S191   3   6K
S197   1   524K
R43   13   50K
R70   811   50K
R85   61   15K
R94   29   1M
R97   22   35.8K
R118   43   37K
R123   11   8K
R165   65   15K
R173   11   6K
R178   4   13K
R185   1   66.3K
R186   36   13K