A Sierpinski/Riesel-like problem

[sweety439]

Originally Posted by sweety439

No (probable) primes found for R93 and R117, they are likely tested to n=8K.

Reserve R85 and R115.

Found these (probable) primes:

(64*85^1253-1)/21
(23*115^1116-1)/2
(51*115^2736-1)/2

[sweety439]

Originally Posted by sweety439

Found these (probable) primes:

(64*85^1253-1)/21
(23*115^1116-1)/2
(51*115^2736-1)/2

Found two (probable) primes:

(169*85^6939-1)/84
(45*115^5227-1)/2

Other k's are likely tested to n=8K with no (probable) primes found.

Released these bases. (R85 is now a 1k base!!!)

[MisterBitcoin]

(3^8972*119-1)/2 will be proven soon, certificate is processing.

[sweety439]

Originally Posted by MisterBitcoin

(3^8972*119-1)/2 will be proven soon, certificate is processing.

Thanks!!!

Please see the post #552, these numbers are in fact the "most wanted" probable primes for me, especially these three numbers:

(29*17^4904-1)/4
(1*51^4229-1)/50
(11*64^3222+1)/3

Also this number not in the post #552:

(61*25^3104+1)/2

[MisterBitcoin]

Originally Posted by sweety439

Thanks!!!

Please see the post #552, these numbers are in fact the "most wanted" probable primes for me, especially these three numbers:

(29*17^4904-1)/4
(1*51^4229-1)/50
(11*64^3222+1)/3

Also this number not in the post #552:

(61*25^3104+1)/2

I´ll do shortly. I´ve now proven all PRP´s <1,5K decimal digits. (for now)
It would be nice if you can maintain the list of unproven PRP´s found by this search.
I also include some ECM factoring up to B1=100M for the N+1/N-1 tests on bigger PRP´s.

[sweety439]

Conjectures:

If {the smallest prime factor of (k*b^n+1)/gcd(k+1,b-1), where n runs the positive integers such that k*b^n is neither perfect odd power nor of the form 4*m^4} is unbounded above, then there are infinitely many n>=1 such that (k*b^n+1)/gcd(k+1,b-1) is prime.

If {the smallest prime factor of (k*b^n-1)/gcd(k-1,b-1), where n runs the positive integers such that k*b^n is not perfect power} is unbounded above, then there are infinitely many n>=1 such that (k*b^n-1)/gcd(k-1,b-1) is prime.

[sweety439]

Originally Posted by sweety439

Conjectures:

If {the smallest prime factor of (k*b^n+1)/gcd(k+1,b-1), where n runs the positive integers such that k*b^n is neither perfect odd power nor of the form 4*m^4} is unbounded above, then there are infinitely many n>=1 such that (k*b^n+1)/gcd(k+1,b-1) is prime.

If {the smallest prime factor of (k*b^n-1)/gcd(k-1,b-1), where n runs the positive integers such that k*b^n is not perfect power} is unbounded above, then there are infinitely many n>=1 such that (k*b^n-1)/gcd(k-1,b-1) is prime.

Another conjecture: (since if k is a rational power of b, then all numbers of the form k*b^n is perfect power)

If k is a rational power of b, then there are infinitely many n>=1 such that (k*b^n-1)/gcd(k-1,b-1) is prime. (this conjecture is related to the generalized repunit primes, i.e. primes of the form (b^n-1)/(b-1))

Note: The conjecture "If k is a rational power of b, then there are infinitely many n such that (k*b^n+1)/gcd(k+1,b-1) is prime" is not true, a counterexample is b=128 and k=8, there is no n>=1 such that (8*128^n+1)/gcd(8+1,128-1) is prime. (see post #265 for many other counterexamples. All counterexamples are listed in post #265, if the first conjecture in post #578 is true)

[MisterBitcoin]

All PRP´s, excluding this and this from page 1 are now proven to be prime.

edit:
I´ve checked this thread up to page 15, the following PRP´s are not-proven, yet:


(61*25^3104+1)/2 (working on it)
(29*17^4904-1)/4 (working on it)

(3^24761*313-1)/2
(7^15118*367-1)/6
(1*51^4229-1)/50
(2626*6^27871-1)/5
(40636*6^18749-1)/5
(54536*6^24822-1)/5

[MisterBitcoin]

The following PRP´s are now proven:



(61*25^3104+1)/2
(29*17^4904-1)/4


I´ve checked this thread up to page 25, the following PRP´s are not-proven, yet:

(11*64^3222+1)/3
(77*61^3080-1)/4
(19*37^5310+1)/4
(43*61^2788+1)/4 (reserving)

(62*61^3698+1)/3
(10243*3^9731+1)/2
(189*31^5570+1)/10
(319*33^5043+1)/32
(19*37^5310+1)/4
(3^24761*313-1)/2
(7^15118*367-1)/6
(1*51^4229-1)/50
(2626*6^27871-1)/5
(40636*6^18749-1)/5
(54536*6^24822-1)/5
(1654*30^38869-1)/29
(197*7^181761-1)/2

[MisterBitcoin]

The following PRP´s are now proven:


(43*61^2788+1)/4
(77*61^3080-1)/4

I´ve checked this thread; the following PRP´s are not-proven, yet:


(215*16^3373+1)/3 (reserving)
(459*16^3701+1)/5 (reserving)

(10*23^3762+1)/11
(43*93^2994+1)/4
(51*115^2736-1)/2
(3356*10^4584+1)/9
(25*67^2829-1)/6
(19*93^4362+1)/4

(11*64^3222+1)/3
(19*37^5310+1)/4
(23*27^3742-1)/2
(44*1024^1933+1)/3

(43*1024^2290-1)/3

(169*85^6939-1)/84
(45*115^5227-1)/2
(370*8^8300+1)/7
(62*61^3698+1)/3
(10243*3^9731+1)/2
(4*115^4223-1)/3
(311*9^15668+1)/8 = (311*81^7834+1)/8

(189*31^5570+1)/10
(621*3^20820+1)/2
(191*105^5045+1)/8
(27*91^5048-1)/2
(3*107^4900-1)/2
(319*33^5043+1)/32
(133*100^5496-1)/33
(13*103^7010+1)/2.
(19*37^5310+1)/4

(79*121^4545-1)/6
(29*13^10574+1)/6

(11*256^5702+1)/3
(407*33^10961+1)/8
(29*13^10574+1)/6
(3^24761*313-1)/2
(7^15118*367-1)/6
(1*51^4229-1)/50
(2626*6^27871-1)/5
(40636*6^18749-1)/5
(152249*6^25389+1)/5
(45634*6^26606+1)/5
(144509*6^28178+1)/5
(17464*6^29081+1)/5
(93589*6^31991+1)/5
(2626*6^29061-1)/5
(2626*6^38681-1)/5
(101529*6^33532+1)/5
(170199*6^25398+1)/5
(2626*6^27871-1)/5
(54536*6^24822-1)/5
(1654*30^38869-1)/29
(197*7^181761-1)/2


Plus 149 PRP´s from S6. Holy, I just got a heart attack when I saw the amount of PRP´s in it.

Some of then where NOT loaded into factordb, I´ll do it when I start to process them. (before some-else proves them.)

[sweety439]

Originally Posted by MisterBitcoin

The following PRP´s are now proven:


(43*61^2788+1)/4
(77*61^3080-1)/4

I´ve checked this thread; the following PRP´s are not-proven, yet:


(215*16^3373+1)/3 (reserving)
(459*16^3701+1)/5 (reserving)

(10*23^3762+1)/11
(43*93^2994+1)/4
(51*115^2736-1)/2
(3356*10^4584+1)/9
(25*67^2829-1)/6
(19*93^4362+1)/4

(11*64^3222+1)/3
(19*37^5310+1)/4
(23*27^3742-1)/2
(44*1024^1933+1)/3

(43*1024^2290-1)/3

(169*85^6939-1)/84
(45*115^5227-1)/2
(370*8^8300+1)/7
(62*61^3698+1)/3
(10243*3^9731+1)/2
(4*115^4223-1)/3
(311*9^15668+1)/8 = (311*81^7834+1)/8

(189*31^5570+1)/10
(621*3^20820+1)/2
(191*105^5045+1)/8
(27*91^5048-1)/2
(3*107^4900-1)/2
(319*33^5043+1)/32
(133*100^5496-1)/33
(13*103^7010+1)/2.
(19*37^5310+1)/4

(79*121^4545-1)/6
(29*13^10574+1)/6

(11*256^5702+1)/3
(407*33^10961+1)/8
(29*13^10574+1)/6
(3^24761*313-1)/2
(7^15118*367-1)/6
(1*51^4229-1)/50
(2626*6^27871-1)/5
(40636*6^18749-1)/5
(152249*6^25389+1)/5
(45634*6^26606+1)/5
(144509*6^28178+1)/5
(17464*6^29081+1)/5
(93589*6^31991+1)/5
(2626*6^29061-1)/5
(2626*6^38681-1)/5
(101529*6^33532+1)/5
(170199*6^25398+1)/5
(2626*6^27871-1)/5
(54536*6^24822-1)/5
(1654*30^38869-1)/29
(197*7^181761-1)/2


Plus 149 PRP´s from S6. Holy, I just got a heart attack when I saw the amount of PRP´s in it.

Some of then where NOT loaded into factordb, I´ll do it when I start to process them. (before some-else proves them.)

(44*1024^1933+1)/3 equals (11*64^3222+1)/3, they are just the same prime from different bases.

Also, you missed these PRPs, if these PRPs were proven, then some bases will be proven.

(11*75^3071+1)/2
(1*91^4421-1)/90

If you prove the primility of (11*75^3071+1)/2, then you will prove S75. Besides, if you prove the primility of both (1*91^4421-1)/90 and (27*91^5048-1)/2, then you will prove R91.