k=1 thru k=12

[gd_barnes]

Originally Posted by sweety439

I don't include n=0 in the searches. In these files, n=0 means no such prime exists, e.g. in the k=12 file, since 12*14^n+1 is always divisible by 13, no primes are of the form 12*14^n+1, and 12*142^n+1 is always divisible by 11 or 13, no primes are of the form 12*142^n+1, and 12*296^n+1 is always divisible by 7, 11, 13, or 19, no primes are of the form 12*296^n+1. Thus, in the file, the n of k=14, 27, 40, ..., 142, ..., 296, etc. are 0.

Originally Posted by sweety439

Thank you, gd_barnes. However, your file still miss some primes, such as 4*737^269302+1 and 4*72^1119849-1. Besides, you only tested 2<=k<=7 for bases 2<=b<=1030, but I want the list of the primes with n>1000 (you only listed n>5000) for 2<=k<=12, 2<=b<=1728 = 12^3 and remain bases for 2<=k<=12 and 2<=b<=1728 = 12^3. I am curious that why you stopped at b=1030.
I only want the k's <= 12, I will not have you give the list of the primes and remain bases for k=13, 14, 15, 16, ..., you don't need to search beyond k=12.

Originally Posted by sweety439

See http://mersenneforum.org/showthread.php?t=15188 for bases 1031<=b<=2048.

For Sierp k=8, all perfect cube bases and all bases =(1 mod 3), =(20 mod 21) should not be searched.
For Riesel k=8, all perfect cube bases and all bases =(1 mod 7), =(20 mod 21) should not be searched.
For Sierp k=9, all bases =(1 mod 2) and all bases =(1 mod 5) should not be searched.
For Riesel k=9, all perfect square bases and all bases =(1 mod 2) and all bases =(4 mod 5) should not be searched.
For Sierp k=10, all bases =(1 mod 11) and all bases =(32 mod 33) should not be searched.
For Riesel k=10, all bases =(1 mod 3) and all bases =(32 mod 33) should not be searched.
For Sierp k=11, all bases =(1 mod 2) and all bases =(1 mod 3) and all bases =(14 mod 15) should not be searched.
For Riesel k=11, all bases =(1 mod 2) and all bases =(1 mod 5) and all bases =(14 mod 15) should not be searched.
For Sierp k=12, all bases =(1 mod 13) and all bases =(142 mod 143) should not be searched.
For Riesel k=12, all bases =(1 mod 11) and all bases =(142 mod 143) should not be searched.

Also, for Riesel k=4, all bases =(4 mod 5) can be proven composite for partial algebraic factors and should not be searched.

Nice research effort Sweety. It now makes sense what you mean for the bases where n=0. I have corrected my post #27 to show the partial algebraic factors for Riesel k=4 and added the primes for Riesel and Sierp k=4. Fortunately these were just posting omissions on my part. They did not affect which bases were remaining for each k so the previously attached files are still correct.

I found all of my files of primes where n<=5000 for these efforts. They are not sorted quite like you want but you should be able to get them into your desired format. They will be attached to the next post.

[gd_barnes]

Attached are all primes for k=2 thru k=7 for bases<=1030 where n<=5000. These are sorted by n-value. Combining these primes with all previously posted primes for n>5000 in post #27 should give all known primes for these k and base ranges.

[sweety439]

There is another problem, the reversed Sierpinski/Riesel problem.

For fixed k, find the smallest base b such that all numbers of the form k*b^n+1 (k*b^n-1) are composite.

If k is of the form 2^n-1 (2^n+1), except k=1 (k=9), it is conjectured for every nontrivial base b, there is a prime of the form k*b^n+1 (k*b^n-1).

However, for all other k's, there is a base b such that all numbers of the form k*b^n+1 (k*b^n-1) are composite.

For the Sierpinski (k*b^n+1) cases: (for k up to 24, only tested bases b<=1030)

S = conjectured smallest base b such that k is a Sierpinski number.

k S remaining bases b with no known primes
1 8 proven
2 201446503145165177 (?) {218, 236, 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004, ...}
3 none {718, 912, ...}
4 14 proven
5 140324348 {308, 326, 512, 824, ...}
6 34 proven
7 none {1004, ...}
8 8 proven
9 177744 {244, 592, 724, 884, 974, 1004, ...}
10 32 proven
11 14 proven
12 142 {12}
13 20 proven
14 38 proven
15 none {398, 650, 734, 874, 876, 1014, ...}
16 38 {32}
17 278 {68, 218}
18 322 {18, 74, 227, 239, 293}
19 14 proven
20 56 proven
21 54 proven
22 68 {22}
23 32 proven
24 114 {79}

[sweety439]

This is I found primes for Sierpinski k=10 with base b<=1030. I didn't search very far, so there are many bases remaining.

I take Sierpinski k=10 because of https://oeis.org/A088622 https://oeis.org/A088782 and https://oeis.org/A088783.

The smallest prime obtained as the concatenation of a power of b followed by a 1, is in fact the smallest prime of the form 10*b^n+1. Thus, this is the reverse Sierpinski problem with k=10.

A088783 lists the bases for which 10 is a Sierpinski number or a trivial k (gcd(10+1, b-1) is not 1), but only up to 166, since when this sequence was created, 10 is still a remaining k for S173. Recently, a prime 10*173^264234+1 was found, next term of this sequence should be 177. However, the correct value of the term after 177 is still unknown, since 10 is a remaining k for S185 (at n=1M).

[gd_barnes]

Originally Posted by sweety439

There is another problem, the reversed Sierpinski/Riesel problem.

For fixed k, find the smallest base b such that all numbers of the form k*b^n+1 (k*b^n-1) are composite.

If k is of the form 2^n-1 (2^n+1), except k=1 (k=9), it is conjectured for every nontrivial base b, there is a prime of the form k*b^n+1 (k*b^n-1).

However, for all other k's, there is a base b such that all numbers of the form k*b^n+1 (k*b^n-1) are composite.

For the Sierpinski (k*b^n+1) cases: (for k up to 24, only tested bases b<=1030)

S = conjectured smallest base b such that k is a Sierpinski number.

k S remaining bases b with no known primes
1 8 proven
2 201446503145165177 (?) {218, 236, 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004, ...}
3 none {718, 912, ...}
4 14 proven
5 140324348 {308, 326, 512, 824, ...}
6 34 proven
7 none {1004, ...}
8 8 proven
9 177744 {244, 592, 724, 884, 974, 1004, ...}
10 32 proven
11 14 proven
12 142 {12}
13 20 proven
14 38 proven
15 none {398, 650, 734, 874, 876, 1014, ...}
16 38 {32}
17 278 {68, 218}
18 322 {18, 74, 227, 239, 293}
19 14 proven
20 56 proven
21 54 proven
22 68 {22}
23 32 proven
24 114 {79}

Interesting list. One thing that I would suggest that CRUS does: Do not include GFN's in your conjectures or bases remaining since the only primes found for them would have to be of the form b^(2^m)+1 where m>=0. Excluding GFNs would change your list as follows:

1. k=1 is not applicable because all odd b have a trivial factor of 2 and all even b are GFNs.
2. For k=2, b=512 is removed since 2*512^n+1=2^(9n+1)+1.
3. For k=12, b=12 is removed since 12*12^n+1=12^(n+1)+1; k=12 is proven.
4. For k=16, b=32 is removed since 16*32^n+1=2^(5n+4)+1; k=16 is proven.
5. For k=18, b=18 is removed since 18*18^n+1=18^(n+1)+1.
6. For k=22, b=22 is removed since 22*12^n+1=22^(n+1)+1; k=22 is proven.

One question: How did you determine the conjectures for k=2, k=5, and k=9?

[gd_barnes]

Originally Posted by sweety439

This is I found primes for Sierpinski k=10 with base b<=1030. I didn't search very far, so there are many bases remaining.

I take Sierpinski k=10 because of https://oeis.org/A088622 https://oeis.org/A088782 and https://oeis.org/A088783.

The smallest prime obtained as the concatenation of a power of b followed by a 1, is in fact the smallest prime of the form 10*b^n+1. Thus, this is the reverse Sierpinski problem with k=10.

A088783 lists the bases for which 10 is a Sierpinski number or a trivial k (gcd(10+1, b-1) is not 1), but only up to 166, since when this sequence was created, 10 is still a remaining k for S173. Recently, a prime 10*173^264234+1 was found, next term of this sequence should be 177. However, the correct value of the term after 177 is still unknown, since 10 is a remaining k for S185 (at n=1M).

Here are some CRUS primes to help fill in your k=10 file:

10*460^751+1
10*708^17562+1
10*830^436+1
10*927^4752+1
10*954^1506+1
10*1012^426+1

[sweety439]

There is an OEIS sequence for the reverse-Sierpinski problem https://oeis.org/A263500.
However, no OEIS sequence for the reverse-Riesel problem, this is my research of this problem:

R = conjectured smallest base b such that k is a Riesel number.

k R remaining bases b with no known primes
1 none (for all base b>2, 1 is a trivial k, and for base b=2, there is a prime with n=2: 1*2^2-1)
2 none
3 none
4 9 proven
5 none
6 24 proven
7 ? (I found no information of this term, can someone find it?)
8 20 proven
9 4 proven
10 32 proven
11 14 proven
12 142 {65, 98}
13 20 proven
14 8 proven
15 ? (I found no information of this term, can someone find it?)
16 9 proven
17 none
18 50 proven
19 14 proven
20 56 proven
21 54 proven
22 68 {38, 62}
23 32 proven
24 114 {64}

There are also OEIS sequence for Sierpinski/Riesel problem:

Sierpinski problem: https://oeis.org/A123159
Riesel problem: https://oeis.org/A273987

[sweety439]

Thanks for some k=10 prime.

About one month ago, I take my effort to find k=10 prime for b=269, 278, 282, 284, 356.

There are some bases b<=1030 remain for k=10:

185 (1M)
338 (100K)
417 (400K)
432 ?
449 ?
537 ?
614 ?
668 ?
671 ?
726 ?
728 ?
743 (200K)
744 (100K)
773 (200K)
786 ?
827 ?
863 ?
869 ?
885 ?
929 ?
935 (200K)
959 ?
977 (100K)
986 ?
1000 (GFN, searched up to (2^25-1)/3-1)
1004 ?

These bases are either not started or have a conjectured k<10.

[sweety439]

For the remaining bases for the reverse-Riesel problem for k=2, 3, 5, 7, 15 and 17:

k R remaining bases b with no known primes
2 none {303, 522, 578, 581, 992, 1019, ...}
3 none {588, 972, ...}
5 none {338, 998, ...}
7 ? {308, 392, 398, 518, 548, 638, 662, 848, 878, ...}
15 ? {454, 552, 734, 856, ...}
17 none {98, 556, 650, 662, 734, ...}

[gd_barnes]

Originally Posted by sweety439

There is an OEIS sequence for the reverse-Sierpinski problem https://oeis.org/A263500.
However, no OEIS sequence for the reverse-Riesel problem, this is my research of this problem:

R = conjectured smallest base b such that k is a Riesel number.

k R remaining bases b with no known primes
1 none (for all base b>2, 1 is a trivial k, and for base b=2, there is a prime with n=2: 1*2^2-1)
2 none
3 none
4 9 proven
5 none
6 24 proven
7 ? (I found no information of this term, can someone find it?)
8 20 proven
9 4 proven
10 32 proven
11 14 proven
12 142 {65, 98}
13 20 proven
14 8 proven
15 ? (I found no information of this term, can someone find it?)
16 9 proven
17 none
18 50 proven
19 14 proven
20 56 proven
21 54 proven
22 68 {38, 62}
23 32 proven
24 114 {64}

There are also OEIS sequence for Sierpinski/Riesel problem:

Sierpinski problem: https://oeis.org/A123159
Riesel problem: https://oeis.org/A273987

I think it is a mistake to include partial or complete algebraic factors for a conjectured k-value especially for Riesel bases. See all of the Riesel bases where k=4 or k=9 would become the conjecture that become uninteresting. On the Sierp side the Sierp OEIS page has a conjecture of k=1 for bases 8 and 32. This is not in the spirit of a good conjecture project. In the case of the reverse conjectures many Riesel k's will have base=4 or 9 as the conjecture leading to many quick proofs that demonstrate little.

Just my two cents.

[sweety439]

According to https://web.archive.org/web/20160507...t/riesel2.html, 3*2^18123-1 is prime, that is, 24*64^3020-1 is prime. Thus, base 64 can be removed for the reverse-Riesel problem with k=24, and the reverse-Riesel problem with k=24 is proven. (This is the smallest prime of the form 24*64^n-1 with n>1, since 18123 is the smallest number greater than 3 and = (3 mod 6) in the k=3 list.