Extended Riesel base conjectures proofs related to the generalized repunit PRP search (i.e. k=1):
Riesel base 184 conjecture of k=36 is proven.
k=4 and 9 were eliminated with algebraic factors.
The largest n's for the k's:
Code:
k n
1 16703
28 85
7 32
16 21
11 15
Riesel base 200 conjecture of k=68 is proven.
For the largest n's for the k's, see CRUS. Since except k=1, all other k's < conjectured k satisfy gcd(k-1,b-1) = 1, thus they are already in CRUS. Thus, the (probable) primes for R200 is just the primes for the same base in CRUS plus the probable prime (200^17807-1)/199 = (1*200^17807-1)/gcd(1-1,200-1).
Riesel base 311 conjecture of k=5 is proven.
The largest n's for the k's:
Code:
k n
1 36497
3 10
4 5
2 2
Riesel base 326 conjecture of k=110 has only k=50 remain (at n=400K).
The largest n's for the k's:
Code:
k n
35 174298
1 26713
98 4562
74 4278
59 1500
(for more (probable) prime for R326, see the text file)
Riesel base 331 conjecture of k=165 has about a dozen k's remain, didn't run it.
Riesel base 371 conjecture of k=5 is proven.
The largest n's for the k's:
Code:
k n
1 15527
2 8
3 2
4 1
Below are Riesel bases where the only k remaining at n=10K is k=1 in order to prove the base conjecture.
Code:
b CK largest n for the k: k (n)
152 16 14 (343720)
185 17 10 (6783)
269 4 2 (20)
281 17 14 (122) (k=9 eliminated with algebraic factors)
384 6 5 (2) (k=4 eliminated with algebraic factors)
Although Riesel bases 380, 385 and 394 have k=1 remain, but since they also have other k's remain (see CRUS, R380 have k=38, 50, 63 and 79 (since except k=1, all other k's < conjectured k satisfy gcd(k-1,b-1) = 1, thus they are already in CRUS. Thus, the (probable) primes for R380 is just the primes for the same base in CRUS plus the (probable) prime for k=1 for the same base), and R394 have k=80 and 86 (and possible some k's with gcd(k-1,394-1) != 1, I have tested this base, but I cannot find (probable) prime for k=19, 22, 25, 37 and many other k's with gcd(k-1,394-1) != 1, notice k=4, 9, 49, 64 and 144 were eliminated with algebraic factors), notice CRUS has not tested Riesel base 385, but since the conjectured k for this base is large, this base should have many k's remain), we do not list them in the list.
For Riesel bases 210, 306 and 396, k=1 already has a prime at n=2, thus the primes we want to find are the primes for k=44100 (for R210), 93636 (for R306), and 156816 (for R396), since we do not allow n=0 (and n<0 :)) but allow n=1. However, the conjectured k for these bases are large, thus these bases should have many k's remain, and except k=44100 for R210, the other two k's are above the conjectured k for the corresponding bases, and we found the prime for R210 k=44100: (210^19819-1)/209 = (44100*210^19817-1)/gcd(44100-1,210-1).