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 2016-12-12, 16:42 #1 sweety439     Nov 2016 54468 Posts 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
 2016-12-13, 07:27 #2 LaurV Romulan Interpreter     Jun 2011 Thailand 7·1,361 Posts 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)
2016-12-13, 14:13   #3
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

20C016 Posts

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 ?

 2016-12-13, 18:53 #4 sweety439     Nov 2016 2×1,427 Posts 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
2016-12-13, 18:55   #5
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts

Quote:
 Originally Posted by sweety439 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.

2016-12-13, 19:37   #6
sweety439

Nov 2016

2·1,427 Posts

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
 base-12-digit-1.txt (230 Bytes, 160 views) base-12-digit-5.txt (12.6 KB, 233 views) base-12-digit-7.txt (2.7 KB, 172 views) base-12-digit-E.txt (129 Bytes, 171 views)

Last fiddled with by sweety439 on 2016-12-13 at 19:39

 2016-12-13, 19:38 #7 sweety439     Nov 2016 2·1,427 Posts For base 2 to base 10, there is a research: https://www.rose-hulman.edu/~rickert/Compositeseq/.
 2016-12-14, 07:43 #8 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3·23·137 Posts 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.
2016-12-14, 10:23   #9
gd_barnes

May 2007
Kansas; USA

1038410 Posts

Quote:
 Originally Posted by sweety439 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

 2016-12-14, 10:33 #10 gd_barnes     May 2007 Kansas; USA 24×11×59 Posts 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
2016-12-14, 11:30   #11
sweety439

Nov 2016

2×1,427 Posts

Quote:
 Originally Posted by gd_barnes 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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