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Old 2021-01-06, 08:54   #89
sweety439
 
Nov 2016

22×3×5×47 Posts
Default

Now, we proved the set of minimal primes (start with b+1, which is equivalent to start with b, if b is composite) of base b=10:

Code:
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
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Old 2021-01-06, 08:56   #90
sweety439
 
Nov 2016

22×3×5×47 Posts
Default

There are totally 77 minimal primes (start with 2 digits) in base 10, there are 75 such primes in base 8
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Old 2021-01-06, 10:36   #91
sweety439
 
Nov 2016

282010 Posts
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Quote:
Originally Posted by sweety439 View Post
Proof of base 4:
Proof of base 6:

The possible (first digit,last digit) for an element with >=3 digits in the minimal set of the strings for primes with at least two digits are:

(1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5)

* Case (1,1):

** 11 is prime, and thus the only minimal prime in this family.

* Case (1,5):

** 15 is prime, and thus the only minimal prime in this family.

* Case (2,1):

** 21 is prime, and thus the only minimal prime in this family.

* Case (2,5):

** 25 is prime, and thus the only minimal prime in this family.

* Case (3,1):

** 31 is prime, and thus the only minimal prime in this family.

* Case (3,5):

** 35 is prime, and thus the only minimal prime in this family.

* Case (4,1):

** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)

*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)

**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.

**** The smallest prime of the form 4{0}41 is 40041

* Case (4,5):

** 45 is prime, and thus the only minimal prime in this family.

* Case (5,1):

** 51 is prime, and thus the only minimal prime in this family.

* Case (5,5):

** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)

*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
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Old 2021-01-06, 12:48   #92
sweety439
 
Nov 2016

B0416 Posts
Default

Currently status for bases 13 to 16:

Code:
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9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, C00068F, C000CFD, C000E2D, C000FAF, C004D4D, C00E20D, C00E8CD, C00F68F, C033A0F, C0802CD, C086AAF, C0A00AF, C0AFFFF, C0C086F, C0C0F8F, C0CA00F, C0CC08F, C0D044D, C0F0AFF, C0FF023, C0FFFAF, C33FA0F, C33FAAF, C3CA00F, C3FFCAF, C8002CD, C8200CD, CCC668F, CCCAA8F, CCCC0A9, CCCC3AF, CCCCCA9, CCCDC4B, CE0008D, CE2000D, CE8CCCD, CF000AF, CFF0AAF, CFFF0AF, D0000EB, D0005EB, D000775, D000EDD, D007077, D00DDD9, D00ED0D, D0AAA45, D0AAAA5, D0EDDDD, D19000B, D4404ED, D4440ED, D5BBBBB, DCCCC4B, DD00DD9, DD07077, DD0DD09, DD0DDD9, DD99999, DDD0D09, DDDD0D9, DDDD9E1, DDDDD09, DDDDD99, DE0DDDD, DEEEEEB, E00001B, E0004A1, E000CAB, E00A041, E00BB0B, E00BBBB, E00C80D, E00CCCB, E044DDD, E0AA4A1, E0AAA41, E0BBB0B, E0D444D, E40444D, E4DDD4D, E88CCCD, E8C000D, E8CCCCD, EA04441, EA0A4A1, EBB000D, EBCCCCD, ED0D00D, EEAAA01, EEBBBBB, EEE000B, F0002C3, F002CC3, F003323, F005545, F00F4A5, F033323, F0400A5, F0A5545, F333323, F333F23, F6660AF, F733333, FA00009, FA004A5, FAAAA45, FC6668F, FCC668F, FD00AA5, FEE7777, FF0F263, FF26003, FF3F323, FF5F887, FFAFF45, FFFF263, FFFF379, 2CCCCC63, 30CCA00F, 33333319, 3333FCAF, 3333FFAF, 33FFA00F, 3C00CCAF, 3C00FCAF, 3CF3FF23, 40000441, 40000CAB, 4000DAA1, 400440DD, 400ACCCB, 400CCCAB, 400E44DD, 4040D00D, 404400DD, 40444EDD, 4044D00D, 40ACCCCB, 40DDDDDD, 440000D1, 44000DDD, 4400DD0D, 44E400DD, 4A00004B, 4A0AAAA5, 5000C08D, 52000CCD, 555400A5, 55540A05, 58800007, 58888087, 5A540005, 5C00020D, 5F5400A5, 5F888887, 60006AAF, 600093AF, 600AAAAF, 608CCCCF, 6600686F, 6606866F, 6688AAAF, 7000077D, 70000D5D, 7000707B, 7000707D, 7000740D, 70500D0D, 7070040D, 707007DD, 7070777B, 7077744D, 7077777B, 77007D0D, 7700B44D, 7707000B, 7707D00D, 7770700D, 7770777B, 7777740D, 7777770B, 7777777D, 77777CAB, 7777B887, 778888E7, 788888E7, 79333333, 7ACCCCCB, 7D0000DD, 7D00D0DD, 7DD00D0D, 7DDDDDA9, 80000081, 80000087, 8000E0CD, 80400E4D, 80A0AAA1, 80EC000D, 84000E4D, 8404444D, 84400E4D, 868AAAAF, 86AAAA8F, 8884044D, 88FFFE77, 8C44444D, 8CCCCAAF, 8E40004D, 900000BB, 90000B0B, 90100009, 90800AA1, 93333AAF, 94AAAAA1, 980000A1, 998AAAA1, A00000F9, A0000EEB, A0005A45, A0055545, A00AAA45, A0666669, A0AAA045, A0AAAA45, A0AAE4A1, A0B44441, A4A00005, A6066669, A8AAFFFF, AA055545, AA0AA045, AAA00A45, AAAAA045, B00000AB, B000EEEB, B00EEE0B, B0900081, B0BBBBAB, B7777787, B9000081, B9008001, B9800001, BA00000B, BBBB0ABB, BCCCCCC9, C000004D, C000086F, C0000AFF, C0000E8D, C0000FDD, C00033AF, C0003CAF, C000448D, C000AFFF, C000CF8F, C004444D, C00663AF, C00F00AF, C00FCCAF, C0FFCCAF, C844444D, CC3A000F, CCCCCBED, CCCCCE2D, CCCCD999, CCDCCC4B, CD44444D, CFAF000F, CFFFF023, D00400ED, D004404D, D00777A5, D00E00DD, D0444E0D, D40000ED, D444E00D, D7DDDDDD, DD00D007, DD0D0077, DD0D0707, DDD0040D, DDDDDD19, DDDDDDD1, E0000CCB, E0044441, E00A4AA1, E888820D, E8888CCD, E888C80D, E8AAAAA1, EB00C0CD, EBBC00CD, ECCCCCCB, F00006AF, F00040A5, F00066AF, F06666AF, F0F004A5, F33FFF23, F60006AF, F6AAA0AF, F88888F7, FE777777, FF33F2C3, FF3FFF23, FF588887, FFFF02C3, FFFF5F6F, FFFFF887, FFFFFF79, 10CCCCCAB, 266666603, 333333AAF, 333333F23, 3333FF2C3, 333CCCCAF, 333FFCCAF, 3A000000F, 3FA00000F, 40000048D, 4000004DD, 4000040D1, 40000ACCB, 4000400D1, 4040000DD, 404D0000D, 40A000005, 40E00444D, 40ED0000D, 444E000DD, 444ED000D, 48444444D, 4A0000005, 4AAAAAAA5, 500000C8D, 500000F8D, 50CCCCC8D, 50FFFFF6F, 5AAAAAA45, 5C020000D, 5E444444D, 666666AFF, 70000044D, 70000440D, 700007CCB, 700007D07, 70044000D, 70070007D, 77070007D, 77700040D, 77700070D, 77707044D, 77770000D, 77777777B, 777888887, 7D0DDDDDD, 7DD0000D7, 8008880A1, 800888A01, 800C000ED, 888800087, 88888AF8F, 888CCCCCD, 88CCCCCCD, 8AAAAAFFF, 8AAFFFFFF, 8CECCCCCD, 8CFFFFCFF, 8EC00000D, 900010009, 908A0AAA1, 9800AAAA1, 9B0CCCCC9, A00000669, A00005545, A0000A545, A000FFF45, A0AAAAA8F, A4000004B, A55540005, A5F554005, AA0A0AA45, AA0AAA8FF, AA4000005, AAA0AA8FF, AAAA0A8FF, AAAA0AA8F, ...}
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Old 2021-01-06, 12:55   #93
sweety439
 
Nov 2016

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Proven minimal primes (start with b+1) set: (bases b = 2, 3, 4, 5, 6, 8, 10, 12) (base b = 7 is not proven, but I cannot find other such primes)

Code:
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}
7: {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}
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, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}
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}

Last fiddled with by sweety439 on 2021-01-26 at 16:35
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Old 2021-01-06, 12:57   #94
sweety439
 
Nov 2016

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Let L(b) be the minimal set of the strings for the primes >b in base b

Code:
b   |L(b)|   largest element in L(b)   largest element in L(b) in base b written in decimal
2   1   11   3
3   3   111   13
4   5   221   41
5   22   109313   5^95+8
6   11   40041   5209
7   71   3161   (7^17-5)/2
8   75   42207   (4*8^221+17)/7
10   77   502827   5*10^30+27
12   106   403977   4*12^41+91
(base b = 7 is not proven, but I cannot find other such primes)

Last fiddled with by sweety439 on 2021-01-26 at 16:36
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Old 2021-01-07, 04:34   #96
LaurV
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Thailand

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Haha, quite ingenious putting them in fdb in a product. Hat off! You made me click on the links because I could not imagine how you can put a list of numbers in fdb under a single identifier. Pleasantly surprised.

(this encouragement post should not be taken as a license to restart spamming the forum, I really do appreciate the cleverness of the solution, but that's all )
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Old 2021-01-07, 07:53   #97
sweety439
 
Nov 2016

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Quote:
Originally Posted by LaurV View Post
Haha, quite ingenious putting them in fdb in a product. Hat off! You made me click on the links because I could not imagine how you can put a list of numbers in fdb under a single identifier. Pleasantly surprised.

(this encouragement post should not be taken as a license to restart spamming the forum, I really do appreciate the cleverness of the solution, but that's all )
Now I computed the set for bases 2, 3, 4, 5, 6, 8, 10 and proved that my sets are all complete (by hand), see post #93, in this post, you said that my set in post #1 for bases 7 and 8 are not complete, and then I tried to solved base 8, and found there are only 4 primes not in the list in post #1:

Code:
77774444441
7777777777771
555555555555525
44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447
For the fully proof of base 8, see post #62

For the fully proof of base 10, see posts #84~#88 in page 8

Now I try to proof base 7, 9, 12 (other bases are too difficult to prove by hand, but can still prove by computer program, but I have no such program that can compute the simple families), base 9 may be harder, since this base has many families which can be ruled out to only contain composites (e.g. {1}, 3{1}, 5{1}, 6{1}, 3{8}, {8}5 and even the non-simple family {1}6{1})
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Old 2021-01-07, 14:32   #98
sweety439
 
Nov 2016

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some 2-digits number xy in base b such that the smallest prime of the form xy*b^n+1 is minimal prime (start with b+1) in base b: (they have the property that both x*b^n+1 and y*b^n+1 are ruled out to only contain composites)

b=11:

Code:
12: 1453
16: 3429742097
18: 199
20: 56204873696128495701224278067068097021
36: 397
38: 419
40: 8574355241
42: 463
48: 6922076933111963233808451075889
56: 617
58: 7019
60: 661
62: 683
78: 859
80: 881
82: 9923
86: 947
100: 12101
102: 1123
106: 187785467
108: 191328589
b=13:

Code:
16: 35153
18: 32258887092667
22: 3719
24: 313
28: 4733
34: 443
40: 521
42: 547
46: 599
48: 48*13^6267+1
66: 859
70: 911
72: 937
76: 166973
94: 1223
96: 1249
100: 1301
102: 1327
106: 25476138101289768358558528047514833801370032844988650960096232747
112: 2609385533717873
118: 211474926496367
120: 120*13^1552+1
124: 1613
126: 276823
144: 1873
148: 25013
150: 1951
154: 2003
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Old 2021-01-07, 15:13   #99
sweety439
 
Nov 2016

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some 3-digits number xyz in base b such that the smallest prime of the form xyz*b^n+1 is minimal prime (start with b+1) in base b: (they have the property that all of x*b^n+1, y*b^n+1, z*b^n+1, xy*b^n+1, xz*b^n+1, yz*b^n+1 are ruled out to only contain composites)

base 11:

Code:
158: 210299
168: 27056569
378: 4159
408: 49369
498: 5479
518: 689459
548: 6029
568: 68729
588: 6469
658: 1165687139
708: 7789
898: 103832913046615294450266950864953559038103755928866797157252835645367668431847219
928: 112289
1138: 137699
1148: 124382424230098512904021128149
base 13:

Code:
184: 2393
190: 917093711
196: 2549
202: 75001187
262: 3407
268: 45293
274: 46307
280: 615161
352: 10053473
358: 292031598119
382: 4967
388: 852437
406: 5279
412: 152972717
430: 5591
436: 5669
460: 5981
466: 13309427
[no known prime for S13 k=484, 2B3{0}1 is unsolved family in base 13]
490: 82811
538: 1181987
544: 1195169
574: 97007
580: 7541
616: 8009
622: 8087
652: 531856430093
658: 90710887636643
874: 264712843161629123
880: 148721
886: 11519
892: 11597
952: 12377
958: 2104727
964: 162917
970: 12611
1198: 34216079
1204: 2645189
1210: 15731
1216: 15809
1276: 80067107693
1282: 36615203
1288: 2829737
1294: 16823
1366: 652758153339229918146262170195611899412691250234661073608281535607441327245782154274226163627511856910242680997040239850166408242104102157590110059393064877741749052590786638302327896695517321882076385248095689607714279245947320040491619053213124101066946141643999231501774298529691499984819614064572813909251081765622950846989229615735750095118621279499641063473790506905622553548920945551313745769942678628213020977681299198153663054881959698284804196609230966508441216040759006656791607
1372: 17837
1396: 18149
1402: 12127883118972635782067
1420: 18461
1426: 18539
1444: 18773
1450: 41413451
1474: 19163
1480: 448255157756534441
1498: 34900531513476539
1504: 19553
1552: 20177
1558: 263303
1588: 2321529421116208423755077
1594: 269387
1630: 21191
1636: 21269
1666: 47582627
1672: 21737
1888: 821828298004735653739505427655167983139612951149032024600411489000613133621406169905393832026983244445467247164419724522787738176502646863025015739869100207704101192365412789485747731430993843624803368161051869616583490868130229284719241207489969244826144259775702398961583808046895149627729346180500126934284621986296138708001440861667299169
1894: 24623
1900: 1549888369901
1906: 4187483
1966: 4319303
1972: 333269
1978: 56493659
1984: 25793
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