20161212, 16:42  #1 
Nov 2016
2^{2}×3×7×23 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=b1 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.rosehulman.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 nearrepunit and quasirepunit (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 20161212 at 16:49 
20161213, 07:27  #2 
Romulan Interpreter
Jun 2011
Thailand
8,539 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 \(\frac{10^{n+1}1}310^n\). (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)/310^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 > 
20161213, 14:13  #3  
"Forget I exist"
Jul 2009
Dumbassville
20300_{8} Posts 
Quote:


20161213, 18:53  #4 
Nov 2016
1932_{10} Posts 
The polynomial of 1{5}1 in base 12 is (16*12^n49)/11.
The polynomial of 23{1} in base 12 is (298*12^n1)/11, in this case I found a proven prime (298*12^16761)/11. For other polynomials for the nearrepdigits and quasirepdigits in base 12 (use X and E for 10 and 11): 1{5}: (16*12^n5)/11 1{7}: (18*12^n7)/11 1{E}: 2*12^n1 2{1}: (23*12^n1)/11 2{5}: (27*12^n5)/11 2{7}: (29*12^n7)/11 2{E}: 3*12^n1 3{1}: (34*12^n1)/11 3{5}: (38*12^n5)/11 3{7}: (40*12^n7)/11 3{E}: 4*12^n1 ... {1}5: (12^n+43)/11 {1}7: (12^n+65)/11 {1}E: (12^n+109)/11 {2}1: (2*12^n13)/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^n25)/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^n25)/11 1{4}1: (15*12^n37)/11 1{5}1: (16*12^n49)/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 20170207 at 15:10 
20161213, 18:55  #5 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 

20161213, 19:37  #6 
Nov 2016
11110001100_{2} Posts 
In base 12:
34{1} cannot be prime since it is equal to (441*12^n1)/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^n1, 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) Last fiddled with by sweety439 on 20161213 at 19:39 
20161213, 19:38  #7 
Nov 2016
2^{2}·3·7·23 Posts 
For base 2 to base 10, there is a research: https://www.rosehulman.edu/~rickert/Compositeseq/.

20161214, 07:43  #8 
"Serge"
Mar 2008
Phi(3,3^1118781+1)/3
10001101000001_{2} 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 
20161214, 10:23  #9  
May 2007
Kansas; USA
2791_{16} Posts 
Quote:
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 20161214 at 10:35 

20161214, 10:33  #10 
May 2007
Kansas; USA
7·1,447 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 20161214 at 10:34 
20161214, 11:30  #11  
Nov 2016
1932_{10} Posts 
Quote:
Last fiddled with by sweety439 on 20161214 at 11:31 

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