This project is from the article

http://www.kurims.kyoto-u.ac.jp/EMIS...rs/i61/i61.pdf, this article is about the mixed Sierpinski (base 2) theorem, which is that for

**every** odd k<78557, there is a prime either of the form k*2^n+1 or of the form 2^n+k, we generalized this theorem (may be only conjectures to other bases) to other prime bases (since the dual form for composite bases is more complex when gcd(k,b) > 1 (see thread

https://mersenneforum.org/showthread.php?t=21954), we only consider prime bases), we conjectured that for every k<

the CK for the Sierpinski conjecture base b (for prime b) which is not divisible by b, there is a prime either of the form k*b^n+1 or of the form b^n+k

We can also generalize this problem to the Riesel side, for the classic (base 2) mixed Riesel problem, there is only 7 unsolved k-values: 2293, 196597, 304207, 342847, 344759, 386801, 444637 (and plus this 2 k-values if probable primes cannot be consider as primes: 363343 and 384539) (see thread

https://mersenneforum.org/showthread.php?t=6545), we conjectured that for every k<

the CK for the Riesel conjecture base b (for prime b) which is not divisible by b, there is a prime either of the form k*b^n-1 or of the form |b^n-k|

Note that the weight of b^n+k is the same as that of k*b^n+1, and the weight of |b^n-k| is the same as that of k*b^n-1, if gcd(k,b)=1

S3 and S7 have too many k's remain, for S5, we have these primes:

Code:

5^24+6436
5^36+7528
5^144+10918
5^1505+26798
5^4+29914
5^458+36412
5^3+41738
5^9+44348
5^485+44738
5^12+45748
5^12+51208
5^46+58642
5^12+60394
5^2+62698
5^2+64258
5^10+67612
5^41+67748
5^13+71492
5^74+74632
5^7+76724
5^3+83936
5^21+84284
5^181+90056
5^23+92906
5^4+93484
5^11+105464
5^11+126134
5^1+139196
5^15+152588

thus the only mixed-remain k-value is 31712

S11 and S13 are already proven, for S17, 17^838+244 is prime, thus, the mixed Sierpinski conjecture base 17 is also a theorem.

For the Riesel side, R3 and R7 also have too many k remain, for R5, we have these primes:

Code:

|5^1-3622|
|5^11-4906|
|5^920-23906|
|5^6-26222|
|5^199-35248|
|5^12-52922|
|5^9-63838|
|5^6-64598|
|5^695-71146|
|5^35-76354|
|5^24-109862|
|5^65-127174|
|5^27-131848|
...

the remain k-values are 68132, 81134, 102952, 109238, 134266, ...

R11, R13, and R17 are already proven.