A Sierpinski/Riesel-like problem

[sweety439]

Originally Posted by gd_barnes

Frankly I don't even understand these particular conjectures and how to search them. I've read the first post here and it still does not make sense. You would have to explain it to me in plain words...no math jargon.

Regardless, the main reason that I'm not insterested is that the word file cannot be updated in real time like a web page can. I will not reserve and search anything until you create a web page that you are continously updating and provide a link to it in the first posting here.

Some of these efforts are somewhat interesting. That is why I have done some searching and also Karsten (kar_bon) did some searching too. In order for you to get people to consistently search for you, you will have to learn how to create web pages and also how to use the correct software so that you can explain that software to people. It is not hard. When I started CRUS nearly 10 years ago I knew nothing about creating web pages and I only knew a little about the software that needed to be used for searching. Basically it just came down to me asking people how it is done and really listening when they explained it to me. Learning those things is part of being the administrator of a project.

These problems are finding and proving the smallest k such that (k*b^n+-1)/gcd(k+-1, b-1) (+ for Sierpinski, - for Riesel) is composite for all natural numbers n>=1, for a given natural number base b>=2.

[sweety439]

Originally Posted by gd_barnes

Frankly I don't even understand these particular conjectures and how to search them. I've read the first post here and it still does not make sense. You would have to explain it to me in plain words...no math jargon.

Regardless, the main reason that I'm not insterested is that the word file cannot be updated in real time like a web page can. I will not reserve and search anything until you create a web page that you are continously updating and provide a link to it in the first posting here.

Some of these efforts are somewhat interesting. That is why I have done some searching and also Karsten (kar_bon) did some searching too. In order for you to get people to consistently search for you, you will have to learn how to create web pages and also how to use the correct software so that you can explain that software to people. It is not hard. When I started CRUS nearly 10 years ago I knew nothing about creating web pages and I only knew a little about the software that needed to be used for searching. Basically it just came down to me asking people how it is done and really listening when they explained it to me. Learning those things is part of being the administrator of a project.

You seems can use srsieve to reserve these problems, for (k*b^n+-1)/gcd(k+-1,b-1), we can only use it to sieve the sequence k*b^n+-1 for the primes p that does not divide gcd(k+-1,b-1), and initialized the list of candidates to not include n for which there is some prime p divides gcd(k+-1,b-1) for which p divides (k*b^n+-1)/gcd(k+-1,b-1).

[sweety439]

Originally Posted by sweety439

The proven Sierpinski problems are:

4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 19, 20, 21, 23, 27, 29, 34, 35, 39, 41, 43, 44, 45, 47, 49, 51, 54, 56, 57, 59, 61*, 64*.

The proven Riesel problems are:

4, 5, 8, 9, 10, 11, 12, 13, 14, 16, 17*, 18, 19, 20, 21, 23, 25, 26, 27, 29, 32, 34, 35, 37, 38, 39, 41, 44, 45, 47, 49, 50, 51*, 53, 54, 55, 56, 57, 59, 62, 64.

* In these bases some of the primes found are probable primes that have not been certified.

The non-certified probable primes for these bases are:

S61:

(62*61^3698+1)/3
(43*61^2788+1)/4

S64:

(11*64^3222+1)/3

R17:

(29*17^4904-1)/4

R51:

(1*51^4229-1)/50

There are also Sierpinski problems that are not proven but the weak case (the GFNs and half GFNs are excluded) are proven, there Sierpinski problems are:

12, 18, 32, 37*, 38, 50, 55, 62.

* In these bases some of the primes found are probable primes that have not been certified.

The non-certified probable primes for these bases are:

S37:

(19*37^5310+1)/4

[sweety439]

Certificates of the proven primes for the proven extended Sierpinski/Riesel conjectures: (>=300 digits and not belong to the original Sierpinski/Riesel conjectures, i.e. gcd(k+-1, b-1) is not 1)

S7, k=141, (141*7^1044+1)/2: http://factordb.com/cert.php?id=1100000000887911448
S13, k=11, (11*13^564+1)/12: (proven to be prime by N-1-method)
S16, k=23, (23*16^1074+1)/3: http://factordb.com/cert.php?id=1100000000001707231
S43, k=13, (13*43^580+1)/14: (proven to be prime by N-1-method)
S43, k=9, (9*43^498+1)/2: http://factordb.com/cert.php?id=1100000000899429028
S61, k=23, (23*61^1659+1)/12: http://factordb.com/cert.php?id=1100000000922387835
R4, k=106, (106*4^4553-1)/3: http://factordb.com/cert.php?id=1100000000350048535
R10, k=121, (121*10^483-1)/3: http://factordb.com/cert.php?id=1100000000291649394
R12, k=298, (298*12^1676-1)/11: http://factordb.com/cert.php?id=1100000000800797310
R17, k=13, (13*17^1123-1)/4: http://factordb.com/cert.php?id=1100000000033706286
R26, k=121, (121*26^1509-1)/5: http://factordb.com/cert.php?id=1100000000894500022
R35, k=1, (1*35^313-1)/34: (proven to be prime by N-1-method)
R37, k=5, (5*37^900-1)/4: (proven to be prime by N-1-method)
R39, k=1, (1*39^349-1)/38: (proven to be prime by N-1-method)
R45, k=53, (53*45^582-1)/4: http://factordb.com/cert.php?id=1100000000920998225
R49, k=79, (79*49^212-1)/6: http://factordb.com/cert.php?id=1100000000854476434
R57, k=87, (87*57^242-1)/2: (proven to be prime by N-1-method)

[sweety439]

Corrected the "k's that make a full covering set with all or partial algebraic factors" column: If there are no k's proven composite by this condition, then delete this condition.

Also, the test limit for R61 is not 5K, it is just 4K, I have corrected it.

This is the right word file.

[sweety439]

This is the text file for the conjectured k and the covering set for all bases b<=64.

[sweety439]

Originally Posted by gd_barnes

Frankly I don't even understand these particular conjectures and how to search them. I've read the first post here and it still does not make sense. You would have to explain it to me in plain words...no math jargon.

Regardless, the main reason that I'm not insterested is that the word file cannot be updated in real time like a web page can. I will not reserve and search anything until you create a web page that you are continously updating and provide a link to it in the first posting here.

Some of these efforts are somewhat interesting. That is why I have done some searching and also Karsten (kar_bon) did some searching too. In order for you to get people to consistently search for you, you will have to learn how to create web pages and also how to use the correct software so that you can explain that software to people. It is not hard. When I started CRUS nearly 10 years ago I knew nothing about creating web pages and I only knew a little about the software that needed to be used for searching. Basically it just came down to me asking people how it is done and really listening when they explained it to me. Learning those things is part of being the administrator of a project.

@Gary:

Can you give all primes for R28 (original conjecture, but only for the k's < the CK for extended conjecture)? Or I cannot complete the top 10 primes for this base (extended conjecture). Also, can you also give all primes for SR22, SR46, SR58 and SR63 (original conjecture, but only for the k's < the CK for extended conjecture)? I only tested these bases to n=1000, but in CRUS, there are primes with n>1000 for these bases, but many of them are not top 10 primes in the original conjecture, so I cannot copy the CRUS primes to this project, see the word files. (Since for SR22, SR46, SR58 and SR63, there are too many k's remain at n=1000, thus I only tested to n=1000)

[gd_barnes]

Originally Posted by sweety439

These problems are finding and proving the smallest k such that (k*b^n+-1)/gcd(k+-1, b-1) (+ for Sierpinski, - for Riesel) is composite for all natural numbers n>=1, for a given natural number base b>=2.

I need some examples to understand this. It appears that the divisor would be different for every k on a single base. That doesn't make sense to me. How could it easily be searched?

I need possibly two sets of examples:
1. An example of the divisor for several consecutive k's on a single base.
2. If I'm not understanding it and the divisor is the same for all k's on a single base, I need an example of the divisor for several consecutive bases.

[gd_barnes]

Originally Posted by sweety439

@Gary:

Can you give all primes for R28 (original conjecture, but only for the k's < the CK for extended conjecture)? Or I cannot complete the top 10 primes for this base (extended conjecture). Also, can you also give all primes for SR22, SR46, SR58 and SR63 (original conjecture, but only for the k's < the CK for extended conjecture)? I only tested these bases to n=1000, but in CRUS, there are primes with n>1000 for these bases, but many of them are not top 10 primes in the original conjecture, so I cannot copy the CRUS primes to this project, see the word files. (Since for SR22, SR46, SR58 and SR63, there are too many k's remain at n=1000, thus I only tested to n=1000)

I will do that when you show that you are serious by (1) creating a web page to show all of this info. -and- (2) posting links to the latest software to use for sieving and searching.

Regardless base 63 has a huge conjecture on both sides. The entire primes file would be multiple gigabytes. I will still consider sending base 63 if you will do the above.

[sweety439]

Originally Posted by gd_barnes

I need some examples to understand this. It appears that the divisor would be different for every k on a single base. That doesn't make sense to me. How could it easily be searched?

I need possibly two sets of examples:
1. An example of the divisor for several consecutive k's on a single base.
2. If I'm not understanding it and the divisor is the same for all k's on a single base, I need an example of the divisor for several consecutive bases.

This divisor is always gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel), e.g. for R7, k=197, the divisor is gcd(197-1,7-1) = 2, and for S10, k=269, the divisor is gcd(269+1,10-1) = 9. Besides, for SR3, the divisor of all even k is 1 and the divisor of all odd k is 2.

Thus, for example, for R13, the divisor of k = 1, 2, 3, ... are {12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4, 1, ...}, and for S11, the divisor of k = 1, 2, 3, ... are {2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, ...}.

In fact, this divisor is the largest number that divides k*b^n+-1 (+ for Sierpinski, - for Riesel) for all n. Thus, this divisor is the largest "trivial factor" of k*b^n+-1 (+ for Sierpinski, - for Riesel).

[sweety439]

See page 12 of https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf, you can use srsieve to sieve (a*b^n+c)/d for all fixed integers a, b, c, d.