mersenneforum.org - Dozenal near- and quasi- repunit primes

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Dozenal near- and quasi- repunit primes

Are there any searching for near- and quasi- repunit primes (primes of the form aaa...aaab, abbb...bbb, aaa...aaabc, abbb...bbbc, abccc...ccc, see thread [URL="https://mersenneforum.org/showthread.php?t=19717"]https://mersenneforum.org/showthread.php?t=19717[/URL]) in dozenal (duodecimal)?

There are a lot of such searching in decimal ([URL="https://stdkmd.net/nrr/#factortables_nr"]https://stdkmd.net/nrr/#factortables_nr[/URL] and [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]https://stdkmd.net/nrr/prime/primedifficulty.txt[/URL]), and I finished this searching in dozenal up to n=1000 (decimal 1728)

[QUOTE=sweety439;545220]Are there any searching for near- and quasi- repunit primes (primes of the form aaa...aaab, abbb...bbb, aaa...aaabc, abbb...bbbc, abccc...ccc, see thread [URL="https://mersenneforum.org/showthread.php?t=19717"]https://mersenneforum.org/showthread.php?t=19717[/URL]) in dozenal (duodecimal)?

There are a lot of such searching in decimal ([URL="https://stdkmd.net/nrr/#factortables_nr"]https://stdkmd.net/nrr/#factortables_nr[/URL] and [URL="https://stdkmd.net/nrr/prime/"]https://stdkmd.net/nrr/prime/[/URL]), and I finished this searching in dozenal up to n=1000 (decimal 1728)[/QUOTE]

These forms have no known (probable) primes:

[CODE]
label expression
{1}55 (10^n+3E7)/E
{2}97 (2*10^n+695)/E
{8}77 (8*10^n-107)/E
{E}9E 10^n-21
20{E} 21*10^n-1
22{E} 23*10^n-1
34{1} (309*10^n-1)/E
53{E} 54*10^n-1
89{1} (804*10^n-1)/E
99{1} (8E4*10^n-1)/E
[/CODE]

However, except the first three forms, all other forms cannot contain a prime because:

10^n-21, 21*10^n-1, (309*10^n-1)/E, 54*10^n-1, (804*10^n-1)/E

even n: algebra factors (difference of two squares)
odd n: factor of 11

23*10^n-1

even n: factor of 11
odd n: algebra factors (difference of two squares)

(8E4*10^n-1)/E

covering set {5, 11, 25}

also note that the form 1{5}1, which is (14*10^n-41)/E, can be prime [I]only for[/I] n=1 because

even n: algebra factors (difference of two squares)
odd n: factor of 11

(and this number for n=1 is exactly 11)

Can someone found a prime of the form {1}55 (111...11155), {2}97 (222...22297), {8}77 (888...88877) in dozenal?

I use srsieve to sieve all {a}b and a{b} forms, and type "srsieve -m 429981696 -n 1729 -N 248832 -p 13 -P 429981696 k.txt" and "srfile -G srsieve.out", however, the result text files are now apply to pfgw, these are the programs and the result text files (unfortunately, the folder is too large (>4 MB), even zipped, so I separate it to 3 zip files), can someone help me how to use srsieve to sieve them and make a file that is apply to pfgw?

[QUOTE=sweety439;545284]I use srsieve to sieve all {a}b and a{b} forms, and type "srsieve -m 429981696 -n 1729 -N 248832 -p 13 -P 429981696 k.txt" and "srfile -G srsieve.out", however, the result text files are now apply to pfgw, these are the programs and the result text files (unfortunately, the folder is too large (>4 MB), even zipped, so I separate it to 3 zip files), can someone help me how to use srsieve to sieve them and make a file that is apply to pfgw?[/QUOTE]

You don't want to use -G, or at least it's not the simplest way to accomplish what you're asking. The text files are essentially 'prp' files, I combined them into one abcd file:

srfile -a "t16_b12.txt" "t17_b12.txt" "t19_b12.txt"

which I then ran on pfgw for a few minutes without finding anything. If you need to anything, you could always switch back between g, a, or G switches. Replacing the files with * symbol

srfile -a "t*_b12.txt"

could save you some time.

BTW you should really check [URL="http://www.primenumbers.net/prptop/searchform.php?form=12%5En%2Bk&action=Search"]PRPtop[/URL] before sieving sequences.

[QUOTE=carpetpool;545427]You don't want to use -G, or at least it's not the simplest way to accomplish what you're asking. The text files are essentially 'prp' files, I combined them into one abcd file:

srfile -a "t16_b12.txt" "t17_b12.txt" "t19_b12.txt"

which I then ran on pfgw for a few minutes without finding anything. If you need to anything, you could always switch back between g, a, or G switches. Replacing the files with * symbol

srfile -a "t*_b12.txt"

could save you some time.

BTW you should really check [URL="http://www.primenumbers.net/prptop/searchform.php?form=12%5En%2Bk&action=Search"]PRPtop[/URL] before sieving sequences.[/QUOTE]

Okay, update the zip file of the abcd file

However, when I do the pfgw.exe for this abcd file, it tested all numbers of the form 1*12^n-11 (numbers of the form {E}1 in dozenal) first, I want to test the numbers [B]sorted by exponent[/B] (i.e. test n=1729 of all forms in the sieve file, then n=1730 of all forms in the sieve file, then n=1731 of all forms in the sieve file, etc.), how to do?

I don't know of any utility that does this but srfile:

[CODE]>>> srfile --help
srfile 0.6.17 -- A file utility for srsieve.
Usage: srfile [OPTION ...] <INFILE ...>
-o --output FILE Write sieve to FILE instead of srsieve.out.
-k --known-factors FILE Remove factors in FILE from the sieve.
-d --delete SEQ Delete sequence SEQ from the sieve e.g. -d "254*5^n-1"
-g --newpgen Write sieve to NewPGen format files t*_b*_k*.npg.
-G --prp Write sieve to PRP (sorted by n) files t*_b*.prp.
-w --pfgw Write sieve to pfgw (sorted by n) file sr_b.pfgw.
-a --abcd Write sieve to abcd format file sr_b.abcd.
-Q --subseqs X Print base b^Q subsequence stats for all Q dividing X.
-c --congruence X Print congruence (mod X) information for sequences.
-p --pfactor X Print Prime95 worktodo.ini entries for P-1 factoring.
X is number of PRP tests saved by finding a factor.
-v --verbose Be verbose.
-q --quiet Be quiet.
-h --help Print this help.
INFILE ... Read sieve from INFILE.[/CODE]

The prp file separates the seqeunces by + or - 1 and + or - c is my understanding. Pfgw's 'abcfileformats.txt' file didn't have much helpful information on converting between ABC type files either. If you have a machine with at least 6 threads, you could try using the 3 prp files you initially had, and test each of them with pfgw simultaneuosly. Last I checked pfgw had a multi threading option, but I don't know how fast it is. You will probably get testing done faster this way.

srfile "-w" option sorts by n!
When running pfgw also use an expression like "{number_primes,$a,1}" in the header, see the documentation there.

Do you know what I tell? The cmd.exe prints:

[CODE]
Recognized ABCD Sieve file:
ABCD File
1*12^1735-11 is composite: RES64: [6BDE769573E9DD87] (0.0390s+0.0017s)
1*12^1741-11 is composite: RES64: [B58118C70216C647] (0.0347s+0.0015s)
1*12^1744-11 is composite: RES64: [142A82E3727A17D2] (0.0367s+0.0012s)
1*12^1746-11 is composite: RES64: [7E35AA5FEEC6F6FD] (0.0380s+0.0014s)
1*12^1749-11 is composite: RES64: [97163264F74252C8] (0.0412s+0.0018s)
1*12^1750-11 is composite: RES64: [E83850812A9BEBC8] (0.0401s+0.0014s)
1*12^1753-11 is composite: RES64: [1DB9B8478E2FA7C6] (0.0399s+0.0016s)
1*12^1758-11 is composite: RES64: [5EAE71FA1429FD5D] (0.0392s+0.0034s)
1*12^1762-11 is composite: RES64: [511A3C0E8959CE0E] (0.0395s+0.0013s)
1*12^1767-11 is composite: RES64: [59E005357B6740DD] (0.0445s+0.0014s)
1*12^1773-11 is composite: RES64: [AE3F4045D68FB340] (0.0692s+0.0014s)
1*12^1775-11 is composite: RES64: [2E6AE6083215A94F] (0.0453s+0.0018s)
1*12^1780-11 is composite: RES64: [98D5BD70E37258CC] (0.0377s+0.0015s)
1*12^1783-11 is composite: RES64: [A3923A1E6AA77841] (0.0427s+0.0013s)
1*12^1785-11 is composite: RES64: [9F1B420E350BFFCC] (0.0430s+0.0013s)
1*12^1793-11 is composite: RES64: [DCCA9D5E4B98EB44] (0.0459s+0.0012s)
1*12^1806-11 is composite: RES64: [EE0AD2158FEEFCDC] (0.0386s+0.0039s)
1*12^1810-11 is composite: RES64: [398A1BC7C63F2956] (0.0475s+0.0014s)
1*12^1813-11 is composite: RES64: [03761C9BFB3ABD46] (0.0486s+0.0014s)
1*12^1835-11 is composite: RES64: [F6E11B4B32134874] (0.0378s+0.0014s)
1*12^1840-11 is composite: RES64: [BCFE8C1BAF1D26B2] (0.0553s+0.0016s)
1*12^1869-11 is composite: RES64: [0569E7AD2978EC5C] (0.0560s+0.0042s)
1*12^1870-11 is composite: RES64: [F41DB45A4575B26D] (0.0551s+0.0010s)
1*12^1873-11 is composite: RES64: [E43D492C726AB3C8] (0.0492s+0.0014s)
1*12^1881-11 is composite: RES64: [EC7DB6D650DCAAA8] (0.0504s+0.0014s)
1*12^1895-11 is composite: RES64: [A1168D47291AFB92] (0.0551s+0.0012s)
1*12^1896-11 is composite: RES64: [A91D2899F9E80912] (0.0500s+0.0015s)
1*12^1901-11 is composite: RES64: [3AE6E7644F645918] (0.0537s+0.0014s)
1*12^1905-11 is composite: RES64: [B883EE2E70080C73] (0.0512s+0.0014s)
1*12^1908-11 is composite: RES64: [60959CD3DBDB153E] (0.0496s+0.0013s)
1*12^1911-11 is composite: RES64: [4B58FDB27213AB3B] (0.0497s+0.0011s)
1*12^1921-11 is composite: RES64: [37CEC4EB70D1574A] (0.0483s+0.0014s)
1*12^1926-11 is composite: RES64: [FCB2F1D40487D934] (0.0581s+0.0013s)
[/CODE]

It runs [I]all numbers of the form 1*12^n-11[/I] first, I want that it runs all numbers with n=1729 first, then n=1730, then n=1731, etc. like this: (the numbers in this list is not right, all numbers divisible by 11 should be divided by 11)

[CODE]
1*12^1729+43
1*12^1729+65
1*12^1729+109
2*12^1729-13
2*12^1729+31
2*12^1729+53
2*12^1729+97
3*12^1729-25
3*12^1729+19
3*12^1729+41
3*12^1729+85
4*12^1729-37
4*12^1729+7
4*12^1729+29
4*12^1729+73
5*12^1729-49
5*12^1729+17
5*12^1729+61
6*12^1729-61
6*12^1729-17
6*12^1729+5
6*12^1729+49
7*12^1729-73
7*12^1729-29
7*12^1729+37
8*12^1729-85
8*12^1729-41
8*12^1729-19
8*12^1729+25
9*12^1729-97
9*12^1729-53
9*12^1729-31
9*12^1729+13
10*12^1729-109
10*12^1729-43
10*12^1729+1
1*12^1729-11
1*12^1729-7
1*12^1729-5
16*12^1729-5
18*12^1729-7
2*12^1729-1
23*12^1729-1
27*12^1729-5
29*12^1729-7
3*12^1729-1
34*12^1729-1
38*12^1729-5
40*12^1729-7
4*12^1729-1
45*12^1729-1
49*12^1729-5
51*12^1729-7
5*12^1729-1
56*12^1729-1
62*12^1729-7
6*12^1729-1
67*12^1729-1
71*12^1729-5
73*12^1729-7
7*12^1729-1
78*12^1729-1
82*12^1729-5
8*12^1729-1
89*12^1729-1
93*12^1729-5
95*12^1729-7
9*12^1729-1
100*12^1729-1
104*12^1729-5
106*12^1729-7
10*12^1729-1
111*12^1729-1
117*12^1729-7
11*12^1729-1
122*12^1729-1
126*12^1729-5
128*12^1729-7
1*12^1730+43
1*12^1730+65
1*12^1730+109
2*12^1730-13
2*12^1730+31
2*12^1730+53
2*12^1730+97
3*12^1730-25
3*12^1730+19
3*12^1730+41
3*12^1730+85
4*12^1730-37
4*12^1730+7
4*12^1730+29
4*12^1730+73
5*12^1730-49
5*12^1730+17
5*12^1730+61
6*12^1730-61
6*12^1730-17
6*12^1730+5
6*12^1730+49
7*12^1730-73
7*12^1730-29
7*12^1730+37
8*12^1730-85
8*12^1730-41
8*12^1730-19
8*12^1730+25
9*12^1730-97
9*12^1730-53
9*12^1730-31
9*12^1730+13
10*12^1730-109
10*12^1730-43
10*12^1730+1
1*12^1730-11
1*12^1730-7
1*12^1730-5
16*12^1730-5
18*12^1730-7
2*12^1730-1
23*12^1730-1
27*12^1730-5
29*12^1730-7
3*12^1730-1
34*12^1730-1
38*12^1730-5
40*12^1730-7
4*12^1730-1
45*12^1730-1
49*12^1730-5
51*12^1730-7
5*12^1730-1
56*12^1730-1
62*12^1730-7
6*12^1730-1
67*12^1730-1
71*12^1730-5
73*12^1730-7
7*12^1730-1
78*12^1730-1
82*12^1730-5
8*12^1730-1
89*12^1730-1
93*12^1730-5
95*12^1730-7
9*12^1730-1
100*12^1730-1
104*12^1730-5
106*12^1730-7
10*12^1730-1
111*12^1730-1
117*12^1730-7
11*12^1730-1
122*12^1730-1
126*12^1730-5
128*12^1730-7
1*12^1731+43
1*12^1731+65
1*12^1731+109
2*12^1731-13
2*12^1731+31
2*12^1731+53
2*12^1731+97
3*12^1731-25
3*12^1731+19
3*12^1731+41
3*12^1731+85
4*12^1731-37
4*12^1731+7
4*12^1731+29
4*12^1731+73
5*12^1731-49
5*12^1731+17
5*12^1731+61
6*12^1731-61
6*12^1731-17
6*12^1731+5
6*12^1731+49
7*12^1731-73
7*12^1731-29
7*12^1731+37
8*12^1731-85
8*12^1731-41
8*12^1731-19
8*12^1731+25
9*12^1731-97
9*12^1731-53
9*12^1731-31
9*12^1731+13
10*12^1731-109
10*12^1731-43
10*12^1731+1
1*12^1731-11
1*12^1731-7
1*12^1731-5
16*12^1731-5
18*12^1731-7
2*12^1731-1
23*12^1731-1
27*12^1731-5
29*12^1731-7
3*12^1731-1
34*12^1731-1
38*12^1731-5
40*12^1731-7
4*12^1731-1
45*12^1731-1
49*12^1731-5
51*12^1731-7
5*12^1731-1
56*12^1731-1
62*12^1731-7
6*12^1731-1
67*12^1731-1
71*12^1731-5
73*12^1731-7
7*12^1731-1
78*12^1731-1
82*12^1731-5
8*12^1731-1
89*12^1731-1
93*12^1731-5
95*12^1731-7
9*12^1731-1
100*12^1731-1
104*12^1731-5
106*12^1731-7
10*12^1731-1
111*12^1731-1
117*12^1731-7
11*12^1731-1
122*12^1731-1
126*12^1731-5
128*12^1731-7
[/CODE]

Also, the abcd file has the number 1*12^n+43, 1*12^65, 1*12^109, etc. instead of (1*12^n+43)/11, (1*12^n+65)/11, (1*12^n+109)/11, etc. however, I want to test the primility for the (1*12^n+43)/11, (1*12^n+65)/11, (1*12^n+109)/11, etc. how to change the numbers in the abcd file? For text file (for the prp file), I know how to do, like my reserving for extended SR46 and SR58: (For every base (b) for the forms (k*b^n+1)/gcd(k+1,b-1) and (k*b^n-1)/gcd(k-1,b-1), there exists a unique value of k for each form that has been conjectured to be the lowest 'Sierpinski value' (+1 form) or 'Riesel value' (-1 form) that is composite for all values of n >= 1. k's make a full covering set with all or partial algebraic factors (e.g. (4*19^n-1)/3, 4*24^n-1, (4*25^n-1)/3, 9*4^n-1, (9*25^n-1)/8, 9*36^n-1) are not considered)

Besides, I use srsieve to sieve them, but srsieve can only sieve a*b^n+c with a>=1, b>=2, c != 0, gcd(a,c)=1, gcd(b,c)=1, it cannot sieve the general case (a*b^n+c)/d (like many cases of dozenal near-repunits:

[CODE]
{1}5 (12^n+43)/11
{1}7 (12^n+65)/11
{1}E (12^n+109)/11
2{1} (23*12^n-1)/11
3{1} (34*12^n-1)/11
4{1} (45*12^n-1)/11
[/CODE]

Thus I sieve 12^n+43, 12^n+65, 12^n+109, 23*12^n-1, 34*12^n-1, 45*12^n-1, etc. instead of (12^n+43)/11, (12^n+65)/11, (12^n+109)/11, (23*12^n-1)/11, (34*12^n-1)/11, (45*12^n-1)/11, etc. in srsieve, but since these numbers are all divisible by 11, thus I can only sieve start with the prime 13, or [I]all numbers[/I] will be removed from the sieve file, and there are still many dozenal near-repunits that are divisible by 5, 7, or 11 in the sieve file, do you know how to remove them?

[QUOTE=sweety439;545626]Besides, I use srsieve to sieve them, but srsieve can only sieve a*b^n+c with a>=1, b>=2, c != 0, gcd(a,c)=1, gcd(b,c)=1, it cannot sieve the general case (a*b^n+c)/d (like many cases of dozenal near-repunits:

[CODE]
{1}5 (12^n+43)/11
{1}7 (12^n+65)/11
{1}E (12^n+109)/11
2{1} (23*12^n-1)/11
3{1} (34*12^n-1)/11
4{1} (45*12^n-1)/11
[/CODE]

Thus I sieve 12^n+43, 12^n+65, 12^n+109, 23*12^n-1, 34*12^n-1, 45*12^n-1, etc. instead of (12^n+43)/11, (12^n+65)/11, (12^n+109)/11, (23*12^n-1)/11, (34*12^n-1)/11, (45*12^n-1)/11, etc. in srsieve, but since these numbers are all divisible by 11, thus I can only sieve start with the prime 13, or [I]all numbers[/I] will be removed from the sieve file, and there are still many dozenal near-repunits that are divisible by 5, 7, or 11 in the sieve file, do you know how to remove them?[/QUOTE]

Srsieve is for numbers of the form k*b^n+-c and sr2sieve requires that k=1 or c=1. Thus, you would have to find a program that sieves your requested forms, or use the -f switch in pfgw instead of a sieve (trial factoring is only slightly slower than actual sieving BTW). Alternatively, you could make a program with GP or some other math library which sieves your specific form. I did this once for other forms that don't have a dedicated sieving program. I could send you an example if you like.

[QUOTE=carpetpool;545650]Srsieve is for numbers of the form k*b^n+-c and sr2sieve requires that k=1 or c=1. Thus, you would have to find a program that sieves your requested forms, or use the -f switch in pfgw instead of a sieve (trial factoring is only slightly slower than actual sieving BTW). Alternatively, you could make a program with GP or some other math library which sieves your specific form. I did this once for other forms that don't have a dedicated sieving program. I could send you an example if you like.[/QUOTE]

You are wrong, I use -w and sorted by n

[CODE]
Recognized ABC Sieve file:
ABC File
2*12^1729-13 is composite: RES64: [FCBEFF5D9726017B] (0.3088s+0.0442s)
3*12^1729+19 is composite: RES64: [ACD8BE7F69CCF93B] (0.2875s+0.1171s)
3*12^1729+41 is composite: RES64: [13D1BC98F11A1A82] (0.8990s+0.1535s)
6*12^1729+49 is composite: RES64: [3AD5B119E2BC4FE2] (0.2647s+0.1758s)
71*12^1729-5 is composite: RES64: [AA02B20400C7D891] (0.7988s+0.1091s)
128*12^1729-7 is composite: RES64: [C8E3BF8CFD691188] (0.3862s+0.1363s)
3*12^1730-1 is composite: RES64: [1CD2CC9E6C8D8C2C] (0.2181s+0.6109s)
3*12^1730+19 is composite: RES64: [8E72C8EF272B1A9B] (0.7866s+0.1368s)
3*12^1730+41 is composite: RES64: [E12318DFFBE36C71] (0.2949s+0.1277s)
5*12^1730-49 is composite: RES64: [F51EBD81224CFD18] (0.2575s+0.1078s)
10*12^1730-1 is composite: RES64: [4DF840F2E104A15E] (0.3002s+0.1506s)
23*12^1730-1 is composite: RES64: [C6CFD00F72C6845B] (0.3583s+0.1165s)
38*12^1730-5 is composite: RES64: [EBA5F05BB4D8C003] (0.3450s+0.1096s)
62*12^1730-7 is composite: RES64: [50DF9889A454B12B] (0.3860s+0.1197s)
73*12^1730-7 is composite: RES64: [C9B26E9494C4DD5A] (0.3311s+0.1635s)
78*12^1730-1 is composite: RES64: [36209BE0322224D6] (0.2912s+0.1105s)
93*12^1730-5 is composite: RES64: [B98A7200C2AABCC3] (0.3199s+0.0997s)
95*12^1730-7 is composite: RES64: [3344A36EFC545CB9] (0.3698s+0.0004s)
[/CODE]

[QUOTE=sweety439;545688]You are wrong, I use -w and sorted by n

[CODE]
Recognized ABC Sieve file:
ABC File
2*12^1729-13 is composite: RES64: [FCBEFF5D9726017B] (0.3088s+0.0442s)
3*12^1729+19 is composite: RES64: [ACD8BE7F69CCF93B] (0.2875s+0.1171s)
3*12^1729+41 is composite: RES64: [13D1BC98F11A1A82] (0.8990s+0.1535s)
6*12^1729+49 is composite: RES64: [3AD5B119E2BC4FE2] (0.2647s+0.1758s)
71*12^1729-5 is composite: RES64: [AA02B20400C7D891] (0.7988s+0.1091s)
128*12^1729-7 is composite: RES64: [C8E3BF8CFD691188] (0.3862s+0.1363s)
3*12^1730-1 is composite: RES64: [1CD2CC9E6C8D8C2C] (0.2181s+0.6109s)
3*12^1730+19 is composite: RES64: [8E72C8EF272B1A9B] (0.7866s+0.1368s)
3*12^1730+41 is composite: RES64: [E12318DFFBE36C71] (0.2949s+0.1277s)
5*12^1730-49 is composite: RES64: [F51EBD81224CFD18] (0.2575s+0.1078s)
10*12^1730-1 is composite: RES64: [4DF840F2E104A15E] (0.3002s+0.1506s)
23*12^1730-1 is composite: RES64: [C6CFD00F72C6845B] (0.3583s+0.1165s)
38*12^1730-5 is composite: RES64: [EBA5F05BB4D8C003] (0.3450s+0.1096s)
62*12^1730-7 is composite: RES64: [50DF9889A454B12B] (0.3860s+0.1197s)
73*12^1730-7 is composite: RES64: [C9B26E9494C4DD5A] (0.3311s+0.1635s)
78*12^1730-1 is composite: RES64: [36209BE0322224D6] (0.2912s+0.1105s)
93*12^1730-5 is composite: RES64: [B98A7200C2AABCC3] (0.3199s+0.0997s)
95*12^1730-7 is composite: RES64: [3344A36EFC545CB9] (0.3698s+0.0004s)
[/CODE][/QUOTE]

WTF.... I forget to divide these numbers by 11

Update the sieve file sorted by exponent. (only for n<=2304, since the original file (n<=12^5) is too large to update here, even when zipped)

[QUOTE=sweety439;545689]WTF.... I forget to divide these numbers by 11[/QUOTE]

How did you sieve them though? I figured you could use the -w option BTW.

[QUOTE=carpetpool;545742]How did you sieve them though? I figured you could use the -w option BTW.[/QUOTE]

I sieved start with the prime 13

[QUOTE=carpetpool;545742]How did you sieve them though? I figured you could use the -w option BTW.[/QUOTE]

For the form (k*12^n+-c)/11, I sieved k*12^n+-c, since srsieve cannot sieve (k*12^n+-c)/11

Update the (probable) primes

[QUOTE=sweety439;545221]These forms have no known (probable) primes:

[CODE]
label expression
{1}55 (10^n+3E7)/E
{2}97 (2*10^n+695)/E
{8}77 (8*10^n-107)/E
{E}9E 10^n-21
20{E} 21*10^n-1
22{E} 23*10^n-1
34{1} (309*10^n-1)/E
53{E} 54*10^n-1
89{1} (804*10^n-1)/E
99{1} (8E4*10^n-1)/E
[/CODE]

However, except the first three forms, all other forms cannot contain a prime because:

10^n-21, 21*10^n-1, (309*10^n-1)/E, 54*10^n-1, (804*10^n-1)/E

even n: algebra factors (difference of two squares)
odd n: factor of 11

23*10^n-1

even n: factor of 11
odd n: algebra factors (difference of two squares)

(8E4*10^n-1)/E

covering set {5, 11, 25}

also note that the form 1{5}1, which is (14*10^n-41)/E, can be prime [I]only for[/I] n=1 because

even n: algebra factors (difference of two squares)
odd n: factor of 11

(and this number for n=1 is exactly 11)

Can someone found a prime of the form {1}55 (111...11155), {2}97 (222...22297), {8}77 (888...88877) in dozenal?[/QUOTE]

Also {3}11 (333...33311) (3×10^n−201)/E, no known (probable) primes for n>2

[QUOTE=sweety439;545798]Also {3}11 (333...33311) (3×10^n−201)/E, no known (probable) primes for n>2[/QUOTE]

Also {1}87 (111...11187) (10^n+6X5)/E

Besides, I found that {3}11 (333...33311) (3×10^n−201)/E cannot be prime since

* For even n, such numbers are divisible by 11
* For odd n, such numbers can be factored as (let n=2*k+1):

((6*10^k-15)/E) * (6*10^k+15)

i.e.

666...6665 * 6000...00015

thus cannot be prime.

update the file of current status (currently at n=8132)

done to n=10007, update current status

I can do better: when written in base 2, [U][B]all[/B][/U] mersenne prime's exponents end in 1.

[QUOTE=LaurV;558296]I can do better: when written in base 2, [U][B]all[/B][/U] mersenne prime's exponents end in 1.[/QUOTE]
All but the first...

[QUOTE=tuckerkao;558291]5 out of the 6 largest known Mersenne Prime exponents are ended with 5 when written in the dozenal base. Also, I cannot find any 9s in all of them.

Ӿ,ӾƐ3,855
12,531,515
17,476,435
20,Ӿ28041
21,Ӿ46,Ɛ85
23,7ӾƐ,125[/QUOTE]

In dozenal, no primes end with 9, since all numbers end with 0, 3, 6, 9 are divisible by 3 (see [URL="https://dozenal.fandom.com/wiki/Divisibility_rule"]Dozenal divisibility rule[/URL])

Also, these project is for the [B]near-repunit and quasi-repunit[/B] primes in dozenal, not for the Mersenne Prime exponents in dozenal.

[URL="https://dozenal.fandom.com/wiki/Near-repdigit_prime"]status for dozenal near-repdigit primes[/URL]

[QUOTE=sweety439;558303]In dozenal, no primes end with 9, since all numbers end with 0, 3, 6, 9 are divisible by 3 (see [URL="https://dozenal.fandom.com/wiki/Divisibility_rule"]Dozenal divisibility rule[/URL])

Also, these project is for the [B]near-repunit and quasi-repunit[/B] primes in dozenal, not for the Mersenne Prime exponents in dozenal.[/QUOTE]
I was mentioning about no 9s for the entire numbers not only the ending units.

For example 9 dozen 1 and 9 dozen 5 are both primes.

[QUOTE=LaurV;558296]I can do better: when written in base 2, [U][B]all[/B][/U] mersenne prime's exponents end in 1.[/QUOTE]
The 0 enders = even numbers, the 1 enders = odd numbers which sound very familiar to everyone.

Base 4 will give more insights as whether the prime exponents turn out to be the 1 ender or the 3 ender.

[QUOTE=tuckerkao;558310]I was mentioning about no 9s for the entire numbers not only the ending units.

For example 9 dozen 1 and 9 dozen 5 are both primes.

The 0 enders = even numbers, the 1 enders = odd numbers which sound very familiar to everyone.

Base 4 will give more insights as whether the prime exponents turn out to be the 1 ender or the 3 ender.[/QUOTE]

Well, there is a list for all Mersenne primes and all Mersenne exponents in dozenal: [URL="https://dozenal.fandom.com/wiki/Mersenne_prime"]https://dozenal.fandom.com/wiki/Mersenne_prime[/URL]

All Mersenne primes > 3 end with 7, and all Mersenne primes > 7 end with either 27 or X7 (27 and X7 are the only two-digit Mersenne primes).
Also, Mersenne exponents end with E are fewer than Mersenne exponents end with 1, 5, or 7, since if p end with E and 2p+1 is also prime (e.g. p = E, 1E, 6E, XE), then Mp is divisible by 2p+1, thus composite.

[QUOTE=Dr Sardonicus;558298]All but the first...[/QUOTE]
Yet, I did better than him! :razz:

[QUOTE=sweety439;558318]Well, there is a list for all Mersenne primes and all Mersenne exponents in dozenal: [URL="https://dozenal.fandom.com/wiki/Mersenne_prime"]https://dozenal.fandom.com/wiki/Mersenne_prime[/URL]

All Mersenne primes > 3 end with 7, and all Mersenne primes > 7 end with either 27 or X7 (27 and X7 are the only two-digit Mersenne primes).
Also, Mersenne exponents end with E are fewer than Mersenne exponents end with 1, 5, or 7, since if p end with E and 2p+1 is also prime (e.g. p = E, 1E, 6E, XE), then Mp is divisible by 2p+1, thus composite.[/QUOTE]
Thanks for the list, it seems like when the exponents end in dozenal 5, it has the slightly higher chance for being a Mersenne Prime.

I have my list for the exponents in dozenal enders, Red for 1, Blue for 5, Pink for 7, Skyblue for Ɛ.