A Sierpinski/Riesel-like problem

[sweety439]

Originally Posted by sweety439

Completed extended R36 for k<=10000 (tested to n=1000).

The remain k's <= 10000 for R36 are: (include the k's without from testing)

251, 260, 924, 1148, 1356, 1555, 1923, 2110, 2133, 2443, 2753, 2776, 3181, 3590, 3699, 3826, 3942, 4241, 4330, 4551, 4635, 4737, 4865, 5027, 5196, 5339, 5483, 5581, 5615, 5791, 5853, 6069, 6236, 6542, 6581, 6873, 6883, 7101, 7253, 7316, 7362, 7399, 7445, 7617, 7631, 7991, 8250, 8259, 8321, 8361, 8363, 8472, 8696, 9036, 9140, 9156, 9201, 9360, 9469, 9491, 9582

[sweety439]

Originally Posted by sweety439

The remain k's <= 10000 for R36 are: (include the k's without from testing)

251, 260, 924, 1148, 1356, 1555, 1923, 2110, 2133, 2443, 2753, 2776, 3181, 3590, 3699, 3826, 3942, 4241, 4330, 4551, 4635, 4737, 4865, 5027, 5196, 5339, 5483, 5581, 5615, 5791, 5853, 6069, 6236, 6542, 6581, 6873, 6883, 7101, 7253, 7316, 7362, 7399, 7445, 7617, 7631, 7991, 8250, 8259, 8321, 8361, 8363, 8472, 8696, 9036, 9140, 9156, 9201, 9360, 9469, 9491, 9582

The k's from the same family are:

{251, 9036}
{260, 9360}

Thus, the remain k<=10000 for R36 are: (totally 59 k's)

251, 260, 924, 1148, 1356, 1555, 1923, 2110, 2133, 2443, 2753, 2776, 3181, 3590, 3699, 3826, 3942, 4241, 4330, 4551, 4635, 4737, 4865, 5027, 5196, 5339, 5483, 5581, 5615, 5791, 5853, 6069, 6236, 6542, 6581, 6873, 6883, 7101, 7253, 7316, 7362, 7399, 7445, 7617, 7631, 7991, 8250, 8259, 8321, 8361, 8363, 8472, 8696, 9140, 9156, 9201, 9469, 9491, 9582

[sweety439]

Originally Posted by kar_bon

No, I thought it was clear, I got my own work: 158 hours of my work here and you're unable to run WinPFGW?

You're testing to n~1500, which is done in seconds with pfgw. I don't know which program you're using so how much can you/we trust your results.

Notes:
- Stop posting tons of files and posts with pages of numbers, update the Wiki pages instead.
- You gave some values in bold, but nowhere explained the meaning.
- Change the display style of the table like this: it's more compact and easier to watch
- Give the GCD for every k-val (I know it's easy to calculate, but noone will do this for many k-values, see example for base 2 & 3)
- Put more own work on current given values instead of creating more new conjectures (2nd, 3rd, 4th CK).

The format of the tables in http://www.mersennewiki.org/index.ph..._definition%29 and http://www.mersennewiki.org/index.ph..._definition%29 are completely the same as that in CRUS, i.e. that in http://www.noprimeleftbehind.net/cru...onjectures.htm and http://www.noprimeleftbehind.net/cru...onjectures.htm.

[sweety439]

Originally Posted by kar_bon

No, I thought it was clear, I got my own work: 158 hours of my work here and you're unable to run WinPFGW?

You're testing to n~1500, which is done in seconds with pfgw. I don't know which program you're using so how much can you/we trust your results.

Notes:
- Stop posting tons of files and posts with pages of numbers, update the Wiki pages instead.
- You gave some values in bold, but nowhere explained the meaning.
- Change the display style of the table like this: it's more compact and easier to watch
- Give the GCD for every k-val (I know it's easy to calculate, but noone will do this for many k-values, see example for base 2 & 3)
- Put more own work on current given values instead of creating more new conjectures (2nd, 3rd, 4th CK).

Oh, I will not create any new conjecture, but I hope that some people will solve all the existing conjectures in http://www.mersennewiki.org/index.ph..._definition%29 and http://www.mersennewiki.org/index.ph..._definition%29, e.g. S10 and S25, thus you can reserve S10 k=269 and S25 k=71.

[sweety439]

The power of 2 bases are extended to base 1024 = 2^10.

These are the text file for the status for S256, S512 and S1024.

Also the text file for all CK for all power of 2 bases b <= 1024.

The CK for all power of 2 bases b <= 1024 are:

base   CK
S2     78557
S4     419
S8     47
S16    38
S32    10
S64    14
S128   44
S256   38
S512   18
S1024  81

base   CK
R2     509203
R4     361
R8     14
R16    100
R32    10
R64    14
R128   44
R256   100
R512   14
R1024  81

The remain k's for these bases are:

base   remain k
S256   11
S512   2, 4, 5, 16
S1024  4, 16, 29, 38, 44, 56

Note:

For S256, all k = 4*m^4 proven composite by full algebra factors.
For S512, all k = m^3 proven composite by full algebra factors.
For S1024, all k = m^5 proven composite by full algebra factors.

The prime (23*256^537+1)/3 (S256, k=23) is converted by S16, k=23.

Some test limits converted by CRUS:

S512, k=5: at n=1M.

Some test limits converted by GFN stats:

S512, k=2: at n=(2^54-1)/9-1
S512, k=4: at n=(2^49-2)/9-1
S512, k=16: at n=(2^44-4)/9-1
S1024, k=4: at n=(2^33-2)/10-1
S1024, k=16: at n=(2^34-4)/10-1

[sweety439]

These are the text files for R256, R512 and R1024.


The remain k's for these bases are:

base   remain k
R256   none (proven)
R512   none (proven)
R1024  13, 29, 31, 43, 56, 61

Note:

For R256, all k = m^2 proven composite by full algebra factors.
For R512, all k = m^3 proven composite by full algebra factors.
For R1024, all k = m^2 and all k = m^5 proven composite by full algebra factors.

The prime 4*512^2215-1 (R512, k=4) is given by CRUS.
The prime 13*512^2119-1 (R512, k=13) is given by CRUS.
The prime 39*1024^4070-1 (R1024, k=39) is given by CRUS.
The prime 74*1024^666084-1 (R1024, k=74) is given by CRUS.

Some test limits converted by CRUS:

R1024, k=29: at n=1M.

[sweety439]

A k is included in the conjecture if and only if this k has infinitely many prime candidates.

Thus, although these k's have a prime, they are excluded from the conjectures:

S8, k=27: Although (27*8^1+1)/gcd(27+1,8-1) is prime, but (27*8^n+1)/gcd(27+1,8-1) is prime only for n=1, because of the algebra factors, thus k=27 is excluded from S8.

S16, k=4: Although (4*16^1+1)/gcd(4+1,16-1) is prime, but (4*16^n+1)/gcd(4+1,16-1) is prime only for n=1, because of the algebra factors, thus k=4 is excluded from S16.

R4, k=1: Although (1*4^2-1)/gcd(1-1,4-1) is prime, but (1*4^n-1)/gcd(1-1,4-1) is prime only for n=2, because of the algebra factors, thus k=1 is excluded from R4.

R4, k=4: Although (4*4^1-1)/gcd(4-1,4-1) is prime, but (4*4^n-1)/gcd(4-1,4-1) is prime only for n=1, because of the algebra factors, thus k=4 is excluded from R4.

R8, k=1: Although (1*8^3-1)/gcd(1-1,8-1) is prime, but (1*8^n-1)/gcd(1-1,8-1) is prime only for n=3, because of the algebra factors, thus k=1 is excluded from R8.

R8, k=8: Although (8*8^2-1)/gcd(8-1,8-1) is prime, but (8*8^n-1)/gcd(8-1,8-1) is prime only for n=2, because of the algebra factors, thus k=8 is excluded from R8.

R8, k=64: Although (64*8^1-1)/gcd(64-1,8-1) is prime, but (64*8^n-1)/gcd(64-1,8-1) is prime only for n=1, because of the algebra factors, thus k=64 is excluded from R8.

R16, k=1: Although (1*16^2-1)/gcd(1-1,16-1) is prime, but (1*16^n-1)/gcd(1-1,16-1) is prime only for n=2, because of the algebra factors, thus k=1 is excluded from R16.

R16, k=16: Although (16*16^1-1)/gcd(16-1,16-1) is prime, but (16*16^n-1)/gcd(16-1,16-1) is prime only for n=1, because of the algebra factors, thus k=16 is excluded from R16.

etc.

[sweety439]

Non-certified probable prime exists only if gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not 1.

[sweety439]

If gcd(k+-1,b-1) = 1 (+ for Sierpinski - for Riesel), then the prime for this k and this base b for this problem (the extended Sierpinski/Riesel problem) is the same as the prime for this k and this base b for the original Sierpinski/Riesel problem (the problem in CRUS).

[sweety439]

In the Riesel case, if k=1, then this problem is completely the same as finding the smallest generalized repunit prime in base b (b should be a non-perfect power, or it would have algebra factors).

For more information for this problem (finding the smallest generalized repunit prime in base b), see http://oeis.org/A084740 and the thread http://mersenneforum.org/showthread.php?t=21808.

[sweety439]

SR108 were done, tested to n=1000, only tested the k's not in CRUS (i.e. k's such that gcd(k+-1,b-1) is not 1).

The remain k for S108 with k = 106 mod 107 are {8987, 14444, 18831, 20543, 21613}

The remain k for R108 with k = 1 mod 107 are {3532, 5351, 6528, 13162}