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Old 2016-12-12, 16:42   #1
sweety439
 
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Default Add repeated digits after a number until it is prime

In base 10, if we add the digit d=1 after the numbers until reach a prime, the research are in http://www.worldofnumbers.com/em197.htm and http://www.worldofnumbers.com/Append...s%20to%20n.txt.

In base b, if we add the digit d=b-1 after the numbers until reach a prime, then the problem is equivalent to the Riesel problem base b (see http://www.noprimeleftbehind.net/cru...onjectures.htm)

See https://www.rose-hulman.edu/~rickert/Compositeseq/ for researched for other bases, for conjectured smallest k such that the numbers (kddd...ddd) are always composite.

Base 2:
d=1: conjectured k = 509202 (1111100010100010010), 50 k's remain.

Base 3:
d=1: conjectured k = 6059 (22022102), 15 k's remain.
d=2: conjectured k = 63064644937 (20000210002020220021021).

Base 4:
d=1: conjectured k = 5 (11), it is proven.
d=3: conjectured k = 8 (20), it is proven.

Base 6:
d=1: conjectured k = 26789 (324005), 29 k's remain.
d=5: conjectured k = 84686 (1452022), only k=1596 remain.

Base 8:
d=1: conjectured k = 21 (25), it is proven.
d=3: conjectured k = 1079770 (4074732), 721 k's remain.
d=5: conjectured k = 7476 (16464), 29 k's remain.
d=7: conjectured k = 13 (15), it is proven.

Base 9:
d=1: conjectured k = 1 (1), it is proven.
d=2: conjectured k = 4615 (6287), 14 k's remain.
d=4: conjectured k = 6059 (8272), 12 k's remain.
d=5: conjectured k = 78 (86), it is proven.
d=7: conjectured k = 2 (2), it is proven.
d=8: conjectured k = 3 (3), it is proven.

Base 10:
d=1: conjectured k = 37 (37), it is proven.
d=3: conjectured k = 4070 (4070), only k=817 remain.
d=7: conjectured k = 891 (891), it is proven.
d=9: conjectured k = 10175 (10175), only k=4420 remain.

Base 12:
d=1: conjectured k = 40 (34), it is proven.
d=5: conjectured k = ? (?) (for remain k's, the first k I cannot find a prime is k=1331)
d=7: conjectured k = 410 (2X2), it is proven.
d=E: conjectured k = 24 (20), it is proven.

I also want to know whether someone is searching near-repunit and quasi-repunit (probable) primes in other bases. (such as 1{5}1 (i.e. 1555...5551) in base 12, I cannot find a prime with this form, or 23{1} (i.e. 23111...111) in base 12, I found a large proven prime 23{1_1676})

Last fiddled with by sweety439 on 2016-12-12 at 16:49
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Old 2016-12-13, 07:27   #2
LaurV
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A nice exercise for you would be to prove that \(38\cdot 10^{3n}+\frac{10^{3n}-1}9\) (i.e. 38'111'111'111'...111) is always divisible by \(\frac{10^{n+1}-1}3-10^n\). (i.e. 2333.....333)

(with a bit of editing, for formatting the text)
Code:
gp > n=0;
gp > [n++, m=(38*10^(3*n)+(10^(3*n)-1)/9)/((10^(n+1)-1)/3-10^n), factorint(m)]
[ 1, 1657, Mat([1657, 1])]
[ 2, 163567, Mat([163567, 1])]
[ 3, 16335667, [31, 1; 526957, 1]]
[ 4, 1633356667, Mat([1633356667, 1])]
[ 5, 163333566667, Mat([163333566667, 1])]
[ 6, 16333335666667, [2857, 1; 5716953331, 1]]
[ 7, 1633333356666667, [73, 1; 22374429543379, 1]]
[ 8, 163333333566666667, [31, 1; 2254501, 1; 2337021457, 1]]
[ 9, 16333333335666666667, [71671, 1; 5274397, 1; 43207441, 1]]
[10, 1633333333356666666667, [139, 1; 52181443, 1; 225187324171, 1]]
[11, 163333333333566666666667, [97, 1; 77863831, 1; 21625558049581, 1]]
[12, 16333333333335666666666667, [19, 1; 103, 1; 283, 1; 1734907, 1; 16998921361351, 1]]
[13, 1633333333333356666666666667, [2617, 1; 624124315373846643739651, 1]]
[14, 163333333333333566666666666667, [1890877, 1; 104708784487, 1; 824951521633, 1]]
[15, 16333333333333335666666666666667, [73, 1; 1861, 1; 120227991530060695506662839, 1]]
[16, 1633333333333333356666666666666667, [223, 1; 8353, 1; 1425757, 1; 615009703935993642649, 1]]
gp >
(or viceversa, if you like :P)
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Old 2016-12-13, 14:13   #3
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Quote:
Originally Posted by sweety
I also want to know whether someone is searching near-repunit and quasi-repunit (probable) primes in other bases. (such as 1{5}1 (i.e. 1555...5551) in base 12, I cannot find a prime with this form, or 23{1} (i.e. 23111...111) in base 12, I found a large proven prime 23{1_1676})
a string of digits is used to represent a number but can be thought of as a polynomial evaluated at x=b where b is the base desired. what do you know about polynomials ?
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Old 2016-12-13, 18:53   #4
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The polynomial of 1{5}1 in base 12 is (16*12^n-49)/11.
The polynomial of 23{1} in base 12 is (298*12^n-1)/11, in this case I found a proven prime (298*12^1676-1)/11.

For other polynomials for the near-repdigits and quasi-repdigits in base 12 (use X and E for 10 and 11):

1{5}: (16*12^n-5)/11
1{7}: (18*12^n-7)/11
1{E}: 2*12^n-1
2{1}: (23*12^n-1)/11
2{5}: (27*12^n-5)/11
2{7}: (29*12^n-7)/11
2{E}: 3*12^n-1
3{1}: (34*12^n-1)/11
3{5}: (38*12^n-5)/11
3{7}: (40*12^n-7)/11
3{E}: 4*12^n-1
...

{1}5: (12^n+43)/11
{1}7: (12^n+65)/11
{1}E: (12^n+109)/11
{2}1: (2*12^n-13)/11
{2}5: (2*12^n+31)/11
{2}7: (2*12^n+53)/11
{2}E: (2*12^n+97)/11
{3}1: (3*12^n-25)/11
{3}5: (3*12^n+19)/11
{3}7: (3*12^n+41)/11
{3}E: (3*12^n+85)/11
...

1{3}1: (14*12^n-25)/11
1{4}1: (15*12^n-37)/11
1{5}1: (16*12^n-49)/11
5{1}5: (56*12^n+43)/11
5{2}5: (57*12^n+31)/11
5{3}5: (58*12^n+19)/11
7{2}7: (79*12^n+53)/11
7{3}7: (80*12^n+41)/11
7{4}7: (81*12^n+29)/11
E{1}E: (122*12^n+109)/11
E{2}E: (123*12^n+97)/11
E{3}E: (124*12^n+85)/11
...

Note: 7{1}7 is always divisible by 11 (decimal 13). Thus, no primes are of the form 7{1}7 in base 12.

Last fiddled with by sweety439 on 2017-02-07 at 15:10
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Old 2016-12-13, 18:55   #5
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Quote:
Originally Posted by sweety439 View Post
The polynomial of 1{5}1 in base 12 is (16*12^n-49)/11.
The polynomial of 23{1} in base 12 is (298*12^n-1)/11, in this case I found a proven prime (298*12^1676-1)/11.
or it's 1*b^n+5*b^(n-1)+... +1 in base b for some value n.
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Old 2016-12-13, 19:37   #6
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In base 12:

34{1} cannot be prime since it is equal to (441*12^n-1)/11, it has algebra factors if n is even and divisible by 13 if n is odd.

2X2{7} cannot be prime since it has a covering set {5, 13, 29}.

20{E} cannot be prime since it is equal to 25*12^n-1, it has algebra factors if n is even and divisible by 13 if n is odd.

These are files for the smallest n>=1 such that k followed by n digits d in base 12 is prime. (all numbers in these files are written in base 10)
Attached Files
File Type: txt base-12-digit-1.txt (230 Bytes, 49 views)
File Type: txt base-12-digit-5.txt (12.6 KB, 119 views)
File Type: txt base-12-digit-7.txt (2.7 KB, 52 views)
File Type: txt base-12-digit-E.txt (129 Bytes, 50 views)

Last fiddled with by sweety439 on 2016-12-13 at 19:39
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Old 2016-12-13, 19:38   #7
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For base 2 to base 10, there is a research: https://www.rose-hulman.edu/~rickert/Compositeseq/.
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Old 2016-12-14, 07:43   #8
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This page is really old.

Consider second to last data line on the page that reads
Code:
value    prime at   composite to
7018      -         630000
But this sequence is prime at 881309 (this find is dated Jul 2013, so the page is more than 3 yrs old). Furthermore, the limit of search for the last remaining value (4420) has since trippled. If you were to mistakenly attempt to "continue that research", you would have wasted cpu-years on something that is already done.
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Old 2016-12-14, 10:23   #9
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Quote:
Originally Posted by sweety439 View Post
In base 12:

34{1} cannot be prime since it is equal to (441*12^n-1)/11, it has algebra factors if n is even and divisible by 13 if n is odd.

2X2{7} cannot be prime since it has a covering set {5, 13, 29}.

20{E} cannot be prime since it is equal to 25*12^n-1, it has algebra factors if n is even and divisible by 13 if n is odd.

These are files for the smallest n>=1 such that k followed by n digits d in base 12 is prime. (all numbers in these files are written in base 10)
I continued the base 12 digit 5 search using PFGW. I extended it to k=5000 and n=25K. Only 2 k's were remaining. Of course the conjecture is unknown so much more searching will be needed. Regardless here are the primes that I found that were n>3000:

k (n)
1331 (6190)
2098 (6542)
2197 (5457)
2852 (5908)
2891 (4196)

2 k's remaining:
4446
4927

Search range:
k<=5000
n<=25000

This was a trivial effort on one core and it took less than a day. You should learn how to use PFGW so that you can extend the k's searched much further.

Last fiddled with by gd_barnes on 2016-12-14 at 10:35
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Old 2016-12-14, 10:33   #10
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http://www.worldofnumbers.com/em197.htm

This page is also out of date but not badly. I have Emailed the following updates to Patrick De Geest:

"Record delayed primes" as coined by Patrick:

Prime by appending 3's (searching done by me with PFGW to n=50K):
[410][337398]
[817[3>50000]

Prime by appending 9's (searching done by the CRUS project to n=1.69M):
[1342] [929711]
[1802] [945881]
[1934] [951836]
[4420] [9>1690000]

Last fiddled with by gd_barnes on 2016-12-14 at 10:34
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Old 2016-12-14, 11:30   #11
sweety439
 
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Quote:
Originally Posted by gd_barnes View Post
I continued the base 12 digit 5 search using PFGW. I extended it to k=5000 and n=25K. Only 2 k's were remaining. Of course the conjecture is unknown so much more searching will be needed. Regardless here are the primes that I found that were n>3000:

k (n)
1331 (6190)
2098 (6542)
2197 (5457)
2852 (5908)
2891 (4196)

2 k's remaining:
4446
4927

Search range:
k<=5000
n<=25000

This was a trivial effort on one core and it took less than a day. You should learn how to use PFGW so that you can extend the k's searched much further.
Thanks very much!

Last fiddled with by sweety439 on 2016-12-14 at 11:31
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