Problems & Puzzles: Puzzles

Problem 83. Follow-up to Puzzle 178

Richard Chen sent the following nice follow-up to puzzle 178, as his added condition to the original Shallit approach:

Puzzle 178 is finding the minimal set of S for a given set S, such as S = {prime = 1 mod 4}, {prime = 3 mod 4}, {palindromic prime}, etc.

A string a is a subsequence of another string b, if a can be obtained from b by deleting zero or more of the characters in b. For example, 514 is a substring of 251664. The empty string is a subsequence of every string.
Two strings a and b are comparable if either a is a substring of b, or b is a substring of a.
A surprising result from formal language theory is that every set of pairwise incomparable strings is finite. This means that from any set of strings we can find its minimal elements.
A string a in a set of strings S is minimal if whenever b (an element of S) is a substring of a, we have b = a.
This set must be finite!

For example, if our set is the set of prime numbers (written in radix 10), then we get the set {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, and if our set is the set of composite numbers (written in radix 10), then we get the set {4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731}

Besides, if our set is the set of prime numbers written in radix b, then we get these sets:

b, we get the set
2: {10, 11}
3: {2, 10, 111}
4: {2, 3, 11}
5: {2, 3, 10, 111, 401, 414, 14444, 44441}
6: {2, 3, 5, 11, 4401, 4441, 40041}

these are already researched in https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf.

Now, let's consider: if our set is the set of prime numbers > b written in radix b, then we get the sets:

(using A−Z to represent digit values 10 to 35)

b: the set for base b

2: {11}

3: {12, 21, 111}

4: {11, 13, 23, 31, 221}

5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031,
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}

6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

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,
777744444417777777777771555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444
44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444
4444444444447}

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, 555555555551, 5000000000000000000000000000027}

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, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

I have only solved these base, i.e. I have found all such primes (elements in the minimal set of the primes >b in base b) and proved that these are all such primes.

For other bases up to 16, I only found all such primes up to certain limit and some larger such primes, e.g. for base 7, I only searched to the prime 5100000001, and the current set is

{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, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545,
5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101,
531101, 1100021, 33333301, 5100000001, ..., 33333333333333331, ...}

I cannot prove that this set is complete.

Also for base 9, I only searched to the prime 8333333335, and the current set is

{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,
5161111111, 8333333335, ..., 300000000035, ..., 311111111161, ..., 544444444444, ..., 2000000000007, ..., 5700000000001,
..., 5111111111111161, ..., 30000000000000000000051, ..., 56111111111111111111111111111111111111, ...,
766666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666
666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666
666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666
6666666666666666666666666666666662, ..., 30000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000011, ...}

And for base 11, I only searched to the prime 1500000001, and the current set is

{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, 10A, 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, 1101, 11A9, 1305, 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, 5105, 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, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911,
AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1,
13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404,
1A411, 1A709, 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, 70103, 70471, 70583, 70714, 71474, 717A4,
71A09, 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, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA,
A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019,
100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223,
222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA,
41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507,
505577, 522A2A, 525223, 528A2A, 531707, 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, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744,
707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484,
777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 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, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999,
A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666,
A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41,
1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 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, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707,
7707778, 7774004, 7777104, 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, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2,
A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744,
22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553,
5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704,
70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477,
77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008,
77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001,
97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707,
A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006,
A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447,
300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A,
5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044,
704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704,
770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477,
777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077,
999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606,
A99999006, A99999099, 1000007447, 1005000007, 1500000001, ..., 3700000001, ..., 4000000005, ..., 600000A999,
..., A000144444, ..., A900000066, ..., 33333333337, ..., 44444444447, ..., A0000000001, ..., A0014444444, ...,
100000000057, ..., 40000000A0041, ..., A000000014444, ..., A044444444441, ..., A144444444411, ..., 40000000000401,
..., A0000044444441, ..., A00000000444441, ..., 11111111111111111, ..., 14444444444441111, ..., 44444444444444111,
..., 70000000000000004, ..., A1444444444444444, ..., A9999999999999996, ..., 888888888888888883, ...,
1444444444444444444, ..., 7777777777777777771, ..., 4000000000000000A041, ..., 9777777777777777777707,
..., A999999999999999999999, ..., 10000000000000000000747, ..., 3577777777777777777777777, ...,
77700000000000000000000008, ..., A44444444444444444444444441, ..., 1500000000000000000000000007,
..., 40000000000000000000000000041, ..., 440000000000000000000000000001, ...,
999999999999999999999999999999991, ..., 1900000000000000000000000000000000001, ...,
A477777777777777777777777777777777777777777, ..., 444444444444444444444444444444444444444444441, ...,
99777777777777777777777777777777777777777777777777777777777777777, ...,
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000051, ..., 55555555555555555555555555555555555555555555555555555555555
55555555555555555555555555555555555
55555555555555555555555555555555555555555555555555555555555555555552A, ...}

The condensed table for bases b up to 16 is:

 b number of quasi-minimal primes base b base-b form of largest known quasi-minimal prime base b length of largest known quasi-minimal prime base b algebraic ((a×bn+c)/d) form of largest known quasi-minimal prime base b 2 1 11 2 3 3 3 111 3 13 4 5 221 3 41 5 22 109313 96 595+8 6 11 40041 5 5209 7 ≥71 3161 17 (717−5)/2 8 75 42207 221 (4×8221+17)/7 9 ≥144 30115811 1161 3×91160+10 10 77 502827 31 5×1030+27 11 ≥895 51612A 163 (11163−57)/2 12 106 403977 42 4×1241+91 13 ≥2451 8032017111 32021 8×1332020+183 14 ≥596 4D19698 19699 5×1419698−1 15 ≥1151 715597 157 (15157+59)/2 16 ≥1877 DB32234 32235 (206×1632234−11)/15

Q1: Complete my set for bases 7, 9, 11 and prove that they are all complete.
Q2: Find all such primes with length <=1000 for bases 13 through 36.
Q3: Find the set for bases 13 through 36. (This will be a hard problem, e.g. the set for base 23
has a probable prime 9(E^800873), and the set for base 30 has a prime O(T^34205))

On August 2, 2021, Richard Chen added:

See this txt file.

***

Richard Chen wrote on June 2, 2022:
All bases 2<=b<=12 and b=14 are completely solved, except the largest minimal prime in base 11 (5(7^62668), its algebraic form is (57*11^62668-7)/10) is only PRP and not proven prime.
See the attached text file.

***

On Set 13, 2002 Richard Chen wrote:
Now, bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 are completely solved, and bases 11, 16, 22, 30 are also solved if probable primes are allowed

This is the data for known minimal primes in bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 30, 36 and the unsolved families in bases 13, 17, 19, 21, 26, 28, 36

...

Condensed table for bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 30, 36: (the bases 11, 13, 16, 17, 19, 21, 22, 26, 28, 30, 36 data assumes the primality of the probable primes)

b
number of minimal primes base b
base-b form of the largest known minimal prime base b
length of the largest known minimal prime base b
algebraic ((a*b^n+c)/d) form of the largest known minimal prime base b
number of unsolved families in base b
searching limit of length for the unsolved families in base b (if there are different searching limits for the unsolved families in base b, choose the lowest searching limit)

2 1 11 2 3 0 ---

3 3 111 3 13 0 ---

4 5 221 3 41 0 ---

5 22 1(0^93)13 96 5^95+8 0 ---

6 11 40041 5 5209 0 ---

7 71 (3^16)1 17 (7^17−5)/2 0 ---

8 75 (4^220)7 221 (4×8^221+17)/7 0 ---

9 151 3(0^1158)11 1161 3×9^1160+10 0 ---

10 77 5(0^28)27 31 5×10^30+27 0 ---

11 1068 5(7^62668) 62669 (57×11^62668−7)/10 0 ---

12 106 4(0^39)77 42 4×12^41+91 0 ---

13 3195~3197 8(0^32017)111 32021 8×13^32020+183 2 139000

14 650 4(D^19698) 19699 5×14^19698−1 0 ---

15 1284 (7^155)97 157 (15^157+59)/2 0 ---

16 2347 (3^116137)AF 116139 (16^116139+619)/5 0 ---

17 10407~10428 E9(B^44732) 44734 (3963×17^44732−11)/16 21 46000

18 549 C(0^6268)C5 6271 12×18^6270+221 0 ---

19 31400~31435 D17D(0^19750)1 19755 89674×19^19751+1 35 20000

20 3314 G(0^6269)D 6271 16×20^6270+13 0 ---

21 13373~13395 5D(0^19848)1 19851 118×21^19849+1 23 20000

22 8003 B(K^22001)5 22003 (251×22^22002−335)/21 0 ---

24 3409 N00(N^8129)LN 8134 13249×24^8131−49 0 ---

26 25250~25259 (5^19391)6F 19393 (26^19393+179)/5 9 20000

28 25528~25529 O4(O^94535)9 94538 (6092×28^94536−143)/9 1 543000

30 2619 O(T^34205) 34206 25×30^34205−1 0 ---

36 35256~35263 (J^10117)LJ 10119 (19×36^10119+2501)/35 7 20000

***

On Dec 13, 2022, Richard wrote:

b number of minimal primes base b base-b form of the top 10 known minimal prime base b (write "dn" if there are 5 or more (n) consecutive same digits d) length of the top 10 known minimal prime base b algebraic ((a×bn+c)/gcd(a+c,b−1)) form of the top 10 known minimal prime base b number of unsolved families in base b searching limit of length for the unsolved families in base b (if there are different searching limits for the unsolved families in base b, choose the lowest searching limit)
2 1 11 2 3 0
3 3 111
21
12
3
2
2
13
7
5
0
4 5 221
31
23
13
11
3
2
2
2
2
41
13
11
7
5
0
5 22 109313
300031
44441
33331
33001
30301
14444
10103
3101
414
96
6
5
5
5
5
5
5
4
3
595+8
9391
3121
2341
2251
1951
1249
653
401
109
0
6 11 40041
4441
4401
51
45
35
31
25
21
15
5
4
4
2
2
2
2
2
2
2
5209
1033
1009
31
29
23
19
17
13
11
0
7 71 3161
51071
3601
1100021
531101
351101
300053
150001
100121
40054
17
10
8
7
6
6
6
6
6
5
(717−5)/2
36×78+1
(78−47)/2
134471
91631
62819
50459
28813
16871
9643
0
8 75

55025
5550525
5500525
4577
221
15
13
11
9
9
8
7
7
7
(4×8221+17)/7
(5×815−173)/7
813−7
(28669×87−25)/7
(53×88−25)/7
(4×89−25)/7
(5×88−2413)/7
1495381
1474901
(4×87+185)/7
0
9 151 30115811
2768607
763292
56136
102557
302051
819335
7271507
511361
1012507
1161
689
331
38
28
23
22
19
16
15
3×91160+10
(23×9688−511)/8
(31×9330−19)/4
(409×936−1)/8
927+52
3×922+46
922−454
(527×917−511)/8
(41×915+359)/8
914+412
0
10 77 502827
5111
80551
66600049
66000049
60549
22051
5200007
946669
666649
31
12
8
8
8
8
8
7
6
6
5×1030+27
(5×1012−41)/9
(725×106−41)/9
66600049
66000049
6×107+49
22×106+1
5200007
946669
666649
0
11 1068 5762668
5571011
775944
A71358
8522005
507206
51612A
5012657
1012551
326122
62669
1013
761
715
223
208
163
129
128
124
(57×1162668−7)/10
(607×111011−7)/10
(7×11761−367)/10
11715−58
(17×11222−111)/2
(557×11206−7)/10
(11163−57)/2
5×11128+62
11127+56
(178×11122−3)/5
0
12 106 403977
B0279B
B699B
AA051
B00099B
AAA0001
BBBAA1
A00065
44AAA1
BBBB1
42
30
9
8
7
7
6
6
6
5
4×1241+91
11×1229+119
129−313
130×126+1

0
13 3196~3197 95197420
8032017111
C523755C
C1063192
B06540BBA
39062661
1770270317
72022972
93015511
715041
197421
32021
23757
10633
6544
6269
2708
2300
1554
1505
(113×13197420−5)/12
8×1332020+183
(149×1323756+79)/12
1310633−50
11×136543+2012
48×136267+1
267×132705+20
93×132298+2
120×131552+1
(7×131505−79)/12
1 300000
14 650 4D19698
34D708
8D14185
886B
408349
8C793
1879B
6B772B
46309
A593
19699
710
144
87
86
81
81
80
65
60
5×1419698−1
47×14708−1
9×14143−79
(8×1487+31)/13
4×1485+65
(116×1480−129)/13
(21×1480+31)/13
(89×1479−1649)/13
(4×1465−667)/13
(10×1460−101)/13
0
15 1284 715597
E145397
9610408
773CE
759CCE
503317
EB31
6330261
705024B
B70241
157
148
107
75
62
36
32
30
28
27
(15157+59)/2
15148−2558
(66×15106−619)/7
(1575+163)/2
(1562+2413)/2
5×1535+22
(207×1531−11)/14
1398×1527+1
1580×1525+11
172×1525+1
0
16 2347 3116137AF
472785DD
DB32234
D0B17804
5BC3700D
90354291
300F1960AF
201713321
F81517F
FAF106245
116139
72787
32235
17806
3703
3545
1965
1717
1519
1066
(16116139+619)/5
(4×1672787+2291)/15
(206×1632234−11)/15
(3131×1617804−11)/15
(459×163701+1)/5
9×163544+145
769×161962−81
2×161716+801
(233×161518+97)/15
251×161064−187
0
17 10409~10427 B671032E
570513101
E9B44732
D0GD37096
G732072F
15024325D
34716074
B3013077D
9D0103985
1090191F
67105
51313
44734
37099
32074
24328
16076
13080
10401
9022
(11×1767105−2411)/16
92×1751311+1
(3963×1744732−11)/16
(60381×1737096−13)/16
(263×1732073+121)/16
22×1724326+13
(887×1716074−7)/16
190×1713078+13
166×1710399+5
179021+32
18 100000
18 549 C06268C5
H766FH
80298B
C0116F5
HD93
GG0301
CF305
B196B
CCF145
714G7
6271
768
300
119
94
33
32
21
17
16
12×186270+221
18768−37
8×18299+11
12×18118+275
(302×1893−13)/17
304×1831+1
(219×1831−185)/17
(11×1821−1541)/17
(3891×1815−185)/17
(7×1816+2747)/17
0
19 31411~31435 D90730469
4F0498476
2482247
2458867A
9042994G
DB36272
333531088
B26588FG
10227907717
C722667C
73049
49850
48225
45888
42996
36273
31091
26590
22795
22669
256×1973047+9
91×1949848+6
(1948225+44)/9
(1945888+926)/9
9×1942995+16
(245×1936272−11)/18
(20579×1931088−5)/18
(11×1926590+1447)/18
1922794+50566
(223×1922668+83)/18
24 75000
20 3314 G06269D
CD2449
501163AJ
J65505J
JCJ629
E566C7
3A5273
G44799
EC04297
40387404B
6271
2450
1166
658
631
568
529
449
432
392
16×206270+13
(241×202449−13)/19
5×201165+219
20658−7881
393×20629−1
(14×20568−907)/19
(67×20528−143)/19
(16×20449−2809)/19
292×20430+7
4×20391+32091
0
21 13382~13394 40473339G
B9045019E5
HD37414
BD35027B
9903323999H
530606FEK
4329236B
J233046J
9211260D
5D0198481
47336
45023
37415
35029
33244
30609
29238
23306
21128
19851
4×2147335+205
240×2145021+299
(353×2137414−13)/20
(233×2135028−53)/20
198×2133242+4175
(2130609+18455)/4
(83×2129237+157)/20
(19×2123306−5479)/20
(9×2121128−3709)/20
118×2119849+1
12 50000
22 8003 BK220015
738152L
L2385KE7
7959K7
J0767IGGJ
K0760EC1
I626AF
E60496L
L483G3
L0454B63
22003
3817
2388
961
772
764
628
499
485
458
(251×2222002−335)/21
(223817−289)/3
222388−653
(22961+857)/3
19×22771+199779
20×22763+7041
(6×22628−1259)/7
314×22497+21
22485−129
21×22457+5459
0
23 65144~65274 71906733
70D0183989
A77M716359
JL015737H
570140481
L13800B
JII013152E
HJE012455J
9B124090B
L11992D7
19069
18402
16363
15740
14051
13801
13156
12459
12412
11999
(7×2319069−2119)/22
3716×2318399+9
(2762239×2316359−7)/22
458×2315738+17
122×2314049+1
(21×2313801−241)/22
10483×2313153+14
9444×2312456+19
(19×2312411−507)/2
(21×2311999−8×237−13)/22
130 20000
24 3409 N00N8129LN
88N5951
A029518ID
D2698LD
N2644LLN
BC0331B
203137
C7298
D0259KKD
I0241I5
8134
5953
2955
2700
2647
334
315
299
263
244
13249×248131−49
201×245951−1
10×242954+5053
(13×242700+4403)/23
242647−1201
276×24332+11
2×24314+7
(283×24298−7)/23
13×24262+12013
18×24243+437
0
25 133598~133730 BH0198655NB
BMA19799O
D71D019471J
BBEF18506D
G0618466FC
L18245I8
1018157D71J
17018147DDJ
BKF0179607
9DN17957
19870
19802
19476
18510
18470
18247
18162
18152
17964
17959
292×2519868+3711
(3569×2519800+163)/12
207538×2519472+19
(57317×2518507−21)/8
(1601×2518468+923)/4
(7×2518247−711)/8
2518161+207544
32×2518150+8469
7390×2517961+7
(5735×2517957−23)/24
132 20000
26 25255~25259 M0611862BB
J044303KCB
6K233005
LD0209757
720279OL
5193916F
9GDK15920P
M8772P
K04364I5
J4222P
61190
44307
23302
20978
20281
19393
15924
8773
4367
4223
22×2661189+1649
19×2644306+13843
(34×2623301−79)/5
559×2620976+7
(7×2620281+11393)/25
(2619393+179)/5
(32569×2615921+21)/5
(22×268773+53)/25
20×264366+473
(19×264223+131)/25
4 100000
27 102831~102900 16193958
KL17469G
Q017273964
7916474G
IP1622807
1015935HN
LH151957
BF14708M
7H0H13557
A013197F9P
19397
17471
17277
16476
16231
15938
15197
14710
13560
13201
(16×2719396+23)/13
(541×2717470−151)/26
26×2717276+6727
(191×2716475+173)/26
(493×2716230−18043)/26
2715937+482
(563×2715196−277)/26
(301×2714709+167)/26
(144629×2713557−17)/26
10×2713200+11203
69 20000
28 25528~25529 O4O945359
5OA31238F
N624051LR
D0526777D
QO423969
537468P
G01899AN
A14236F
5I1370F
51332P8P
94538
31241
24054
5271
4242
3748
1902
1425
1372
1335
(6092×2894536−143)/9
(4438×2831239+125)/27
(209×2824053+3967)/9
13×285270+5697
(242×284241−4679)/9
(5×283748+2803)/27
16×281901+303
(10×281425−2899)/27
(17×281371−11)/3
(5×281335+426163)/27
1 543202
29
30 2619 OT34205
I024608D
54882J
C010221
M0547SS7
M241QB
AN206
50164B
J153QJ
J94QQJ
34206
24610
4883
1024
551
243
207
166
155
97
25×3034205−1
18×3024609+13
(5×304883+401)/29
12×301023+1
22×30550+26047
(22×30243+3139)/29
(313×30206−23)/29
5×30165+11
(19×30155+6071)/29
(19×3097+188771)/29
0
31
32 168832~169017 V09018867D
T018762F
DM18004L
F17783H
V1775333
A17650I5
8017186MJ
V6171079
V16755O3
C016737AAA9
18871
18764
18006
17784
17755
17652
17189
17109
16757
16742
31753×3218868+13
29×3218763+15
(425×3218005−53)/31
(15×3217784+47)/31
3217755−925
(10×3217652+7771)/31
8×3217188+723
(967×3217108+87)/31
3216757−253
12×3216741+338249
185 20000
33
34 184750~184834 U19778KCF
B195254B
20193714N
H0O18511X
8018215QP
7TTM18060F
B170281
H01700781
N016800DKX
Q1632105
19781
19527
19374
18514
18218
18064
17029
17010
16804
16323
(10×3419781−134067)/11
(3419527−715)/3
2×3419373+159
(6366×3418512+91)/11
8×3418217+909
(27323×3418061−23)/3
(3417029−31)/3
17×3417009+273
23×3416803+15741
(26×3416323−29891)/33
84 20000
35
36 35286~35290 P81993SZ
S0750078H
7K26567Z
J10117LJ
VL07258J
EO06177V
FZ57773P
T0946181
RY4562H
OZ3932AZ

***

 Records   |  Conjectures  |  Problems  |  Puzzles

 Home | Melancholia | Problems & Puzzles | References | News | Personal Page | Puzzlers | Search | Bulletin Board | Chat | Random Link Copyright © 1999-2012 primepuzzles.net. All rights reserved.