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2019-11-29, 09:08   #12
sweety439

Nov 2016

1000101101102 Posts

Quote:
 Originally Posted by LaurV I didn't look yet how good is your code, but my former one is lousy, so there are chances that yours is better. I mean, not the code, but my method itself was lousy, to look at all primes one by one. The authors of that paper you linked describe a method which is much better and somehow similar to what I am doing now. Right now, I split the problem in two steps, first I let the zero apart, and solve the problem with "digits" from 1 to b-1, by starting from the end with all possible cases in a set. Starting from the end or from the beginning makes no difference, but in the case the base is even, I only have n/2 elements in the initial set (because numbers ending in 2, 4, 6, etc, can never be prime), so the search dimension is reduced in half. Then, for all elements in set, I check what digit I can add in front of them and still avoiding conflicts. If any of the resulting numbers is prime, I add it to the set. Here is where the algorithm "strikes", because I can do this in about linear time, by creating a matrix with the possible candidates, and then eliminating them from the matrix, by different criteria (like, it produces conflict, it is a prime and I add it to the list, or it is always composite regardless of how you extend it, etc), and sometimes full rows and columns can be eliminated. This gives me the complete set, excluding the numbers that contain zero, in just minutes. The second part comes from the realization that the numbers that contain zero and have to be in the set, if we delete zeroes from them, the new created are (1) still not in the set, and (2) can not be covered with numbers in the set, and (3) are the same magnitude as the numbers in the set except maybe the first digit, that can repeat indefinitely till the first prime is found. The (3) is very important (and it can be proved) so the second part of the algorithm is to create a list with all such numbers (like 5-6 digit numbers in our case) and see which one becomes a prime when it is "stuffed" with zeroes, which is piece of cake. Mind that the zeros have to be "between" the digits, as "leading zeros do not count"(TM) and numbers ending in zero in any base are not prime.
Unfortunately, my program also look at all primes one by one

Is there a better program to write all minimal prime <= 1000 digits in <= 5 minute? Like that we can use program to write all repunit prime <= 1000 digits in <= 5 minute

Can we take all forms that may have primes? Like https://github.com/curtisbright/mepn...nsolved.25.txt (base 25) and https://github.com/RaymondDevillers/.../master/left31 (base 31), etc.

Last fiddled with by sweety439 on 2019-11-29 at 09:11

 2019-11-29, 14:39 #13 yae9911     "Hugo" Jul 2019 Germany 31 Posts The cited GitHub repositories don't provide the programs to calculate the sets of minimal base-n representations, but the lists themselves are given. See e.g. for n=8:minimal.8.txt From the discussion here I have learned that the length of the lists with the shortest entries are not in the OEIS. That's why I added the corresponding entries A330048 and A330049 With the filling of such defects I am merciless and fast. I could also insert a link to this discussion, or you could provide a PARI program to compute the initial terms of the sequence. No need to be efficient, but demonstrating the principle.
2019-11-30, 00:26   #14
sweety439

Nov 2016

2×5×223 Posts

Quote:
 Originally Posted by yae9911 The cited GitHub repositories don't provide the programs to calculate the sets of minimal base-n representations, but the lists themselves are given. See e.g. for n=8:minimal.8.txt From the discussion here I have learned that the length of the lists with the shortest entries are not in the OEIS. That's why I added the corresponding entries A330048 and A330049 With the filling of such defects I am merciless and fast. I could also insert a link to this discussion, or you could provide a PARI program to compute the initial terms of the sequence. No need to be efficient, but demonstrating the principle.
My problem is not for the set of minimal base-n representations of the primes, it is for the set of minimal base-n representations of the primes >= n, i.e. single-digit primes are not counted.

Thus, e.g. for base 5:

original set is {2, 3, 10, 111, 401, 414, 14444, 44441}
new set is {10, 12, 21, 23, 32, 34, 43, 111, 131, 133, 313, 401, 414, 14444, 30301, 33001, 33331, 44441, 300031}

For base 6:

original set is {2, 3, 5, 11, 4401, 4441, 40041}
new set is {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Last fiddled with by sweety439 on 2019-11-30 at 00:41

2019-11-30, 00:59   #15
sweety439

Nov 2016

2×5×223 Posts

Quote:
 Originally Posted by yae9911 The cited GitHub repositories don't provide the programs to calculate the sets of minimal base-n representations, but the lists themselves are given. See e.g. for n=8:minimal.8.txt From the discussion here I have learned that the length of the lists with the shortest entries are not in the OEIS. That's why I added the corresponding entries A330048 and A330049 With the filling of such defects I am merciless and fast. I could also insert a link to this discussion, or you could provide a PARI program to compute the initial terms of the sequence. No need to be efficient, but demonstrating the principle.
However, https://oeis.org/A326609 is in OEIS, A330049(n) is the length of A326609(n) in base n.

A330048(17) is either 1279 or 1280, A330048(19) is either 3462 or 3463, A330048(21) is either 2599 or 2600, 17597<=A330048(25)<=17609, 5662<=A330048(26)<=5664, also, A330048(30)=220, 6295<=A330048(36)<=6297, 37773<=A330048(40)<=37774 (I found the prime Qa{U12380}X in base 40), A330048(42)=4551, 29103<=A330048(48)<=29109, see https://github.com/RaymondDevillers/primes/

A330049(30)=1024, A330049(42)=487.

Besides, I saw A327282, this is A327282(n) for 28<=n<=48:

Code:
n,A327282(n)
28,131
29,123
30,207
31,147
32,160
33,163
34,201
35,169
36,216
37,173
38,185
39,195
40,242
41,205
42,331
43,229
44,242
45,252
46,277
47,261
48,411
(I only searched up to 4 digits, I assume that there are no minimal composites with >=5 digits in these bases)

Also, all A330048, A330049 and A327282 should have the keyword "base".

Last fiddled with by sweety439 on 2019-11-30 at 01:00

2019-11-30, 06:43   #16
sweety439

Nov 2016

223010 Posts

Quote:
 Originally Posted by sweety439 However, https://oeis.org/A326609 is in OEIS, A330049(n) is the length of A326609(n) in base n. A330048(17) is either 1279 or 1280, A330048(19) is either 3462 or 3463, A330048(21) is either 2599 or 2600, 17597<=A330048(25)<=17609, 5662<=A330048(26)<=5664, also, A330048(30)=220, 6295<=A330048(36)<=6297, 37773<=A330048(40)<=37774 (I found the prime Qa{U12380}X in base 40), A330048(42)=4551, 29103<=A330048(48)<=29109, see https://github.com/RaymondDevillers/primes/ A330049(30)=1024, A330049(42)=487. Besides, I saw A327282, this is A327282(n) for 28<=n<=48: Code: n,A327282(n) 28,131 29,123 30,207 31,147 32,160 33,163 34,201 35,169 36,216 37,173 38,185 39,195 40,242 41,205 42,331 43,229 44,242 45,252 46,277 47,261 48,411 (I only searched up to 4 digits, I assume that there are no minimal composites with >=5 digits in these bases) Also, all A330048, A330049 and A327282 should have the keyword "base".
A327282(n) for 49<=n<=75:

Code:
49,294
50,292
51,290
52,322
53,299
54,438
55,331
56,304
57,331
58,356
59,339
60,659
61,375
62,379
63,404
64,461
65,412
66,613
67,416
68,419
69,449
70,647
71,464
72,696
73,505
74,499
75,538
This is enough to fill the "data" section of A327282

Last fiddled with by sweety439 on 2019-11-30 at 06:44

2019-11-30, 13:42   #17
sweety439

Nov 2016

42668 Posts

Quote:
 Originally Posted by sweety439 .... Now, let's consider: if our set is the set of prime numbers >= b written in radix b (i.e. the prime numbers with at least two digits in radix b), then we get the sets: Code: b, we get the set 2: {10, 11} 3: {10, 12, 21, 111} 4: {11, 13, 23, 31, 221} 5: {10, 12, 21, 23, 32, 34, 43, 111, 131, 133, 313, 401, 414, 14444, 30301, 33001, 33331, 44441, 300031} 6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} 7: {10, 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1112, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 30011, 31111, 33001, 33311, 35555, 40054, 300053, 33333301} 8: {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441} However, I do not think that my base 7 and 8 sets are complete (I use PARI program to find these primes (all written in base b), but I only searched the primes with <= 8 digits, so there may be missing primes), I proved that my base 2, 3, 4, 5 and 6 sets are complete. Can someone complete my base 7 and 8 set? Also find the sets of bases 9 to 36.
For bases 9 to 12:

Code:
b, we get the set
9: {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007}
10: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551}
11: {10, 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 11A9, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A11, 1774A, 17777, 177A4, 17A47, 1A114, 1A411, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70471, 70583, 70714, 71474, 717A4, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 222223, 222823, 300202, 300323, 303203, 307577, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 440A41, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A05509, A0A058, A0A955, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1A44444, 2555505, 2845055, 3030023, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 717444A, 7400404, 7700717, 7707778, 7774004, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A006906, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 22888823, 28888223, 30555777, 33000023}
12: {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001}
Can someone complete them?

Last fiddled with by sweety439 on 2019-11-30 at 13:42

 2019-12-03, 03:58 #18 sweety439     Nov 2016 223010 Posts For base 11, I found these numbers: (for the primes with at least two digits) 10, 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 11A9, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A11, 1774A, 17777, 177A4, 17A47, 1A114, 1A411, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70471, 70583, 70714, 71474, 717A4, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 222223, 222823, 300202, 300323, 303203, 307577, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 440A41, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701777, 704408, 704417, 704457, 704484, 707041, 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