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Old 2017-01-06, 16:37   #45
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Base 2:
d=1: conjectured k = 509202 (1111100010100010010), 52 k's remain at n>=4.8M.

Base 3:
d=1: conjectured k = 6059 (22022102), 15 k's remain at n=50K.
d=2: conjectured k = 63064644937 (20000210002020220021021), 277577 k's remain at n>=25K.

Base 4:
d=1: conjectured k = 120 (1320), it is proven. (k's such that 3*k+1 = m^2 are proven composite by full algebraic factors)
d=3: conjectured k = 39938 (21300002), 7 k's remain at n>=2.4M. (k's such that k+1 = m^2 are proven composite by full algebraic factors)

Base 5:
d=1: conjectured k = 3 (3), it is proven.
d=2: conjectured k = 191115 (22103430), 258 k's remain at n=15K.
d=3: conjectured k = 585655 (122220110), 2274 k's remain at n=15K.
d=4: conjectured k = 346801 (42044201), 74 k's remain at n=2.2M.

Base 6:
d=1: conjectured k = 26789 (324005), 24 k's remain at n=25K.
d=5: conjectured k = 84686 (1452022), only k=1596 remain at n=5M.

Base 7:
d=1: conjectured k = 76 (136), it is proven.
d=2: conjectured k = 15979 (64405), 17 k's remain at n=15K.
d=3: conjectured k = 5629 (22261), 19 k's remain at n=15K.
d=4: conjectured k = 20277 (113055), 16 k's remain at n=15K.
d=5: conjectured k = 43 (61), it is proven.
d=6: conjectured k = 408034255081 (41323316641135), 8391 k's <= 500M remain at n>=25K.

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

Base 9:
d=1: conjectured k = 5 (5), it is proven. (k's such that 8*k+1 = m^2 are proven composite by full algebraic factors)
d=2: conjectured k = 4615 (6287), 14 k's remain at n=15K.
d=4: conjectured k = 6059 (8272), 12 k's remain at n=18K.
d=5: conjectured k = 78 (86), it is proven.
d=7: conjectured k = 2 (2), it is proven.
d=8: conjectured k = 73 (81), it is proven. (k's such that k+1 = m^2 are proven composite by full algebraic factors)

Base 10:
d=1: conjectured k = 37 (37), it is proven.
d=3: conjectured k = 4070 (4070), only k=817 remain at n=554K.
d=7: conjectured k = 891 (891), it is proven.
d=9: conjectured k = 10175 (10175), only k=4420 remain at n=1.7M.

Base 11:
d=1: conjectured k = 3 (3), it is proven.
d=2: conjectured k = 2627 (1X79), 14 k's remain at n=5K.
d=3: conjectured k = 1520 (1162), 22 k's remain at n=5K.
d=4: conjectured k = 973 (805), 6 k's remain at n=5K.
d=5: conjectured k = 2 (2), it is proven.
d=6: conjectured k = 1669093 (X40018), no searching done.
d=7: conjectured k = 3 (3), it is proven.
d=8: conjectured k = 4625 (3525), 25 k's remain at n=5K.
d=9: conjectured k = 716 (5X1), 5 k's remain at n=5K.
d=X: conjectured k = 861 (713), it is proven.

Base 12:
d=1: conjectured k = 117 (99), it is proven. (k's such that 11*k+1 = m^2 and m = 5 or 8 mod 13, or 11*k+1 = 3*m^2 and m = 3 or 10 mod 13 are proven composite by partial algebraic factors)
d=5: conjectured k = 33441 (17429), 31 k's remain at n=25K.
d=7: conjectured k = 410 (2X2), it is proven.
d=E: conjectured k = 375 (273), it is proven. (k's such that k+1 = m^2 and m = 5 or 8 mod 13, or k+1 = 3*m^2 and m = 3 or 10 mod 13 are proven composite by partial algebraic factors)

Last fiddled with by sweety439 on 2017-02-03 at 17:32
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Old 2017-01-06, 19:08   #46
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Quote:
Originally Posted by sweety439 View Post
Base 2:
d=1: conjectured k = 509202 (1111100010100010010), 52 k's remain at n>=4.8M.

Base 3:
d=1: conjectured k = 6059 (22022102), 15 k's remain at n=50K.
d=2: conjectured k = 63064644937 (20000210002020220021021), 277577 k's remain at n>=25K.

Base 4:
d=1: conjectured k = 120 (1320), it is proven. (k's such that 3*k+1 = m^2 are proven composite by full algebraic factors)
d=3: conjectured k = 39938 (21300002), 7 k's remain at n>=2.4M. (k's such that k+1 = m^2 are proven composite by full algebraic factors)

Base 5:
d=1: conjectured k = 3 (3), it is proven.
d=2: conjectured k = 191115 (22103430), 258 k's remain at n=15K.
d=3: conjectured k = 585655 (122220110), 2274 k's remain at n=15K.
d=4: conjectured k = 346801 (42044201), 74 k's remain at n=2.2M.

Base 6:
d=1: conjectured k = 26789 (324005), 24 k's remain at n=25K.
d=5: conjectured k = 84686 (1452022), only k=1596 remain at n=5M.

Base 7:
d=1: conjectured k = 76 (136), it is proven.
d=2: conjectured k = 15979 (64405), 17 k's remain at n=15K.
d=3: conjectured k = 5629 (22261), 19 k's remain at n=15K.
d=4: conjectured k = 20277 (113055), 16 k's remain at n=15K.
d=5: conjectured k = 43 (61), it is proven.
d=6: conjectured k = 408034255081 (41323316641135), 8391 k's <= 500M remain at n>=25K.

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

Base 9:
d=1: conjectured k = 5 (5), it is proven. (k's such that 8*k+1 = m^2 are proven composite by full algebraic factors)
d=2: conjectured k = 4615 (6287), 14 k's remain at n=15K.
d=4: conjectured k = 6059 (8272), 12 k's remain at n=18K.
d=5: conjectured k = 78 (86), it is proven.
d=7: conjectured k = 2 (2), it is proven.
d=8: conjectured k = 73 (81), it is proven. (k's such that k+1 = m^2 are proven composite by full algebraic factors)

Base 10:
d=1: conjectured k = 37 (37), it is proven.
d=3: conjectured k = 4070 (4070), only k=817 remain at n=554K.
d=7: conjectured k = 891 (891), it is proven.
d=9: conjectured k = 10175 (10175), only k=4420 remain at n=1.7M.

Base 11:
d=1: conjectured k = 3 (3), it is proven.
d=2: conjectured k = 2627 (1X79), 14 k's remain at n=5K.
d=3: conjectured k = 1520 (1162), 22 k's remain at n=5K.
d=4: conjectured k = 973 (805), 6 k's remain at n=5K.
d=5: conjectured k = 2 (2), it is proven.
d=6: conjectured k = 1669093 (X40018), no searching done.
d=7: conjectured k = 3 (3), it is proven.
d=8: conjectured k = 4625 (3525), 25 k's remain at n=5K.
d=9: conjectured k = 716 (5X1), 5 k's remain at n=5K.
d=X: conjectured k = 861 (713), it is proven.

Base 12:
d=1: conjectured k = 117 (99), it is proven. (k's such that 11*k+1 = m^2 and m = 5 or 8 mod 13 are proven composite by partial algebraic factors)
d=5: conjectured k = 33441 (17429), 31 k's remain at n=25K.
d=7: conjectured k = 410 (2X2), it is proven.
d=E: conjectured k = 375 (273), it is proven. (k's such that k+1 = m^2 and m = 5 or 8 mod 13 are proven composite by partial algebraic factors)
For the k's proven composite by algebraic factors for base 12:
d=1: k's such that 11*k+1 = m^2 and m = 5 or 8 mod 13, or 11*k+1 = 3*m^2 and m = 3 or 10 mod 13 are proven composite by partial algebraic factors.
d=E: k's such that k+1 = m^2 and m = 5 or 8 mod 13, or k+1 = 3*m^2 and m = 3 or 10 mod 13 are proven composite by partial algebraic factors.

Last fiddled with by sweety439 on 2017-01-06 at 19:09
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Old 2017-01-21, 15:40   #47
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This is the text file for base 10, d=7.

Since in https://www.rose-hulman.edu/~rickert/Compositeseq/, there is no list for base 10, d=7.

The conjectured k is 891, since 891*10^n+7*(10^n-1)/9 has a covering set {3, 11, 13, 37}, period=6. This conjecture is proven.

In this text file, n's > 1000 are given by http://oeis.org/A090464 (711*10^1648+7*(10^1648-1)/9), http://oeis.org/A257460 (337*10^2184+7*(10^2184-1)/9 and 599*10^2508+7*(10^2508-1)/9) and http://www.worldofnumbers.com/em197.htm (95*10^2904+7*(10^2904-1)/9, 480*10^11330+7*(10^11330-1)/9 and 851*10^28895+7*(10^28895-1)/9).
Attached Files
File Type: txt base-10-digit-7.txt (5.2 KB, 57 views)

Last fiddled with by sweety439 on 2017-01-21 at 15:50
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Old 2017-01-22, 11:53   #48
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The proven conjectures are: (*In these cases some of the primes found are probable primes that have not been certified)

Base 4, d=1, k=120.
Base 5, d=1, k=3.
*Base 7, d=1, k=76. (there are 2 non-certified PRPs in this case: 52*7^5907+1*(7^5907-1)/6 and 61*7^15118+1*(7^15118-1)/6)
Base 7, d=5, k=43.
Base 8, d=1, k=21.
Base 8, d=7, k=13.
Base 9, d=1, k=5.
Base 9, d=5, k=78.
Base 9, d=7, k=2.
Base 9, d=8, k=73.
Base 10, d=1, k=37.
*Base 10, d=7, k=891. (there are 2 non-certified PRPs in this case: 480*10^11330+7*(10^11330-1)/9 and 851*10^28895+7*(10^28895-1)/9)
Base 11, d=1, k=3.
Base 11, d=5, k=2.
Base 11, d=7, k=3.
Base 11, d=X, k=861.
Base 12, d=1, k=117.
Base 12, d=7, k=410.
Base 12, d=E, k=375.

Last fiddled with by sweety439 on 2017-01-22 at 11:55
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Old 2017-04-22, 04:48   #49
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I have searched all of the applicable bases <=12 and digits to n=25K with the exception of base 5 digit 3, which was searched to n=10K. Many errors were found in the linked page shown in the first post. It should be ignored. Results are shown below.

References to CRUS are to the Conjectures 'R Us page at http://www.noprimeleftbehind.net/crus. Applicable k's remaining and PRPs for n>5000 are attached.

Base 2:
d=1: conjectured k = 509202; 52 k's remain at n>=4.8M; see CRUS.

Base 3:
d=1: conjectured k = 6059; 15 k's remain at n=50K;
(k=806, 915, 968, 1565, 1794, 2877, 3393, 3738, 3813, 3969, 4356, 4388, 4905, 5325, 5798); see attached PRP list.
d=2: conjectured k = 63064644937; 274148 k's remain at n>=25K; see CRUS.

Base 4:
d=1: conjectured k = 120; proven;
(k=5, 8, 16, 21, 33, 40, 56, 65, 85, 96 proven composite with the help of algebraic factors).
d=3: conjectured k = 39938; 7 k's remain at n>=2.4M;
(k=4585, 9220, 9518, 14360, 19463, 23668, 31858) see CRUS;
(k=8, 35, 80, 143, 224, 323, etc. proven composite with the help of algebraic factors).

Base 5:
d=1; conjectured k = 3; proven.
d=2; conjectured k = 191115; 201 k's remain at n=25K; see attached k's remaining and PRP list.
d=3; conjectured k = 585655; 1843 k's remain at n=10K; see attached k's remaining and PRP list.
d=4; conjectured k = 346801; 74 k's remain at n=2.3M; see CRUS.

Base 6:
d=1: conjectured k = 26789; 24 k's remain at n=25K;
(k=525, 1247, 1898, 7406, 9954, 10137, 10788, 11012, 11585, 15352, 15960, 16682, 16982, 17243, 17759, 18592, 20770, 21868, 22232, 23758, 24683, 25277, 25788, 26495); see attached PRP list.
d=5: conjectured k = 84686; 1 k remains at n=5M; (k=1596); see CRUS.

Base 7:
d=1: conjectured k = 76; proven; some PRP's are n>5K; see attached PRP list.
d=2: conjectured k = 15979; 13 k's remain at n=25K;
(k=2047, 4011, 4101, 4785, 5149, 6479, 6609, 8305, 9331, 9839, 12347, 14057, 15181); see attached PRP list.
d=3: conjectured k = 5629; 22 k's remain at n=25K;
(k=98, 358, 589, 905, 989, 1336, 1945, 2465, 2579, 3329, 3499, 3518, 4229, 4256, 4565, 4579, 4586, 4759, 4828, 5095, 5116, 5546); see attached PRP list.
d=4: conjectured k = 20277; 17 k's remain at n=25K;
(k=507, 3405, 4203, 4469, 6397, 6517, 6739, 8687, 10159, 10397, 10697, 12293, 12635, 14953, 18107, 19675, 19703); see attached PRP list.
d=5: conjectured k = 43; proven.
d=6: conjectured k = 408034255081; only partial search done; see CRUS.

Base 8:
d=1: conjectured k = 21; proven.
d=3: conjectured k = 1079770; 309 k's remain at n=25K; see attached k's remaining and PRP list.
d=5: conjectured k = 7476; 25 k's remain at n=25K;
(k=486, 539, 1791, 1898, 2046, 2274, 2336, 2489, 2982, 3458, 3503, 4121, 4283, 4464, 5292, 5531, 5627, 6267, 6312, 6611, 6851, 7014, 7232, 7233, 7398); see attached PRP list.
d=7: conjectured k = 13; proven; see CRUS.

Base 9:
d=1: conjectured k = 5; proven;
(k=1, 3 proven composite with the help of algebraic factors).
d=2: conjectured k = 4615; 11 k's remain at n=25K;
(k=307, 847, 975, 1157, 1255, 1945, 2995, 3157, 3985, 4167, 4195); see attached PRP list.
d=4: conjectured k = 6059; 11 k's remain at n=25K;
(k=915, 1565, 2419, 2877, 2905, 3393, 3813, 3969, 4905, 5325, 5383); see attached PRP list.
d=5: conjectured k = 78; proven.
d=7: conjectured k = 2; proven.
d=8: conjectured k = 73; proven; see CRUS;
(k=3, 15, 35, 63 proven composite with the help of algebraic factors).

Base 10:
d=1: conjectured k = 37; proven.
d=3: conjectured k = 4070; 1 k remains at n=50K (others may have searched to n=554K); (k=817); see attached PRP list.
d=7: conjectured k = 891; proven; some PRP's are n>5K; see attached PRP list.
d=9: conjectured k = 10175; 1 k remains at n=1.75M; (k=4420); see CRUS.

Base 11:
d=1: conjectured k = 3; proven.
d=2: conjectured k = 2627, 8 k's remain at n=25K;
(k=653, 857, 1067, 1297, 1337, 1781, 2299, 2439); see attached PRP list.
d=3: conjectured k = 1520; 14 k's remain at n=25K;
(k=17, 178, 188, 379, 511, 551, 589, 682, 770, 829, 988, 1073, 1178, 1436); see attached PRP list.
d=4: conjectured k = 973; 3 k's remain at n=25K;
(k=61, 333, 865); see attached PRP list.
d=5: conjectured k = 2; proven.
d=6: conjectured k = 1669093; 434 k's remain at n=25K; see attached k's remaining and PRP list.
d=7: conjectured k = 3; proven.
d=8: conjectured k = 4625; 14 k's remain at n=25K;
(k=57, 131, 533, 609, 743, 1167, 1253, 1769, 1961, 2147, 2231, 4047, 4053, 4523); see attached PRP list.
d=9: conjectured k = 716; 3 k's remain at n=25K;
(k=227, 337, 535); see attached PRP list.
d=A: conjectured k = 861; proven; see CRUS.

Base 12:
d=1: conjectured k = 117; proven;
(k=40, 105 proven composite with the help of algebraic factors).
d=5: conjectured k = 33441; 31 k's remain at n=25K; see attached k's remaining and PRP list.
d=7: conjectured k = 410; proven.
d=B: conjectured k = 375; proven; see CRUS;
(k=24, 26, 63, 299, 323 proven composite with the help of algebraic factors).
Attached Files
File Type: zip prp-repeat-digit.zip (15.3 KB, 66 views)
File Type: zip remain-repeat-digit.zip (9.5 KB, 71 views)

Last fiddled with by sweety439 on 2017-06-28 at 13:55 Reason: correct the remain k's for base 4
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Old 2017-06-11, 21:39   #50
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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:

base 4, d=1, k=1: Although 1*4^1+(4^1-1)/3 is prime, but 1*4^n+(4^n-1)/3 is prime only for n=1, because of the algebra factors, thus k=1 is excluded from b4d1.

base 8, d=1, k=1: Although 1*8^2+(8^2-1)/7 is prime, but 1*8^n+(8^n-1)/7 is prime only for n=2, because of the algebra factors, thus k=1 is excluded from b8d1.

base 8, d=9, k=1: Although 9*8^1+(8^1-1)/7 is prime, but 9*8^n+(8^n-1)/7 is prime only for n=1, because of the algebra factors, thus k=9 is excluded from b8d1.
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Old 2019-07-03, 05:34   #51
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Consider a related problem:

The number of iterations (x -> a*x+b) until a prime is found, starting with n; or 0 if a prime is never found. (where a, b, n are integers, a > 1 (the a = 1 case can be proven by Dirichlet's theorem), b != 0, n > 0, gcd(a,b) = 1, gcd(n,b) = 1)
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Old 2019-07-03, 05:39   #52
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Quote:
Originally Posted by sweety439 View Post
Consider a related problem:

The number of iterations (x -> a*x+b) until a prime is found, starting with n; or 0 if a prime is never found. (where a, b, n are integers, a > 1 (the a = 1 case can be proven by Dirichlet's theorem), b != 0, n > 0, gcd(a,b) = 1, gcd(n,b) = 1)
Code:
(a,b)  conjectured smallest n such that gcd(n,b) = 1 and the iterations (x -> a*x+b) can never be prime
(2,1)   509202
(2,3)
(2,5)
(2,7)
(2,-1)  78558
(2,-3)
(2,-5)
(2,-7)
(3,1)   6059
(3,2)   63064644937
(3,-1)  5524
(3,-2)  125050976087
(4,1)   120
(4,3)   39938
(4,5)
(4,7)
(4,-1)  140
(4,-3)  66742
(4,-5)
(4,-7)

Last fiddled with by sweety439 on 2019-07-03 at 05:40
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Old 2019-07-04, 18:33   #53
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Quote:
Originally Posted by sweety439 View Post
Consider a related problem:

The number of iterations (x -> a*x+b) until a prime is found, starting with n; or 0 if a prime is never found. (where a, b, n are integers, a > 1 (the a = 1 case can be proven by Dirichlet's theorem), b != 0, n > 0, gcd(a,b) = 1, gcd(n,b) = 1)

I wrote a small Mathematica notebook to handle this (see attached zip file). Note I did make one change: if the number n is already prime then the number of iterations is 0.
Feel free to play around with the parameters. The output is a bunch of cells which has three numbers: the starting n, the number of iterations, and the ending prime.
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Old 2019-07-22, 14:15   #54
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Quote:
Originally Posted by Dylan14 View Post
I wrote a small Mathematica notebook to handle this (see attached zip file). Note I did make one change: if the number n is already prime then the number of iterations is 0.
Feel free to play around with the parameters. The output is a bunch of cells which has three numbers: the starting n, the number of iterations, and the ending prime.
You only ran 2*n+1 (and only ran for n<=100), which is equivalent to the Riesel base 2 problem.

I also have the other formulas like 2*n-1 which is equivalent to the Sierpinski base 2 problem), 2*n+-3, 2*n+-5, 2*n+-7, 3*n+-1, 3*n+-2, 3*n+-4, 3*n+-5, 3*n+-7, 3*n+-8, ,4*n+-1, 4*n+-3, 4*n+-5, 4*n+-7, 5*n+-1, 5*n+-2, 5*n+-3, 5*n+-4, 6*n+-1, 6*n+-5, ...
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Old 2019-07-23, 12:12   #55
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These are the text files for 2x+-3 for x<=1024 (gcd(x,3) =1), note that for 2x-3, this x must be >3, or the numbers would be negative.
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File Type: txt 2x+3.txt (4.6 KB, 27 views)
File Type: txt 2x-3.txt (4.6 KB, 26 views)
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