20211212, 03:07  #1 
May 2018
3^{5} Posts 
Primes in centuries: The Tortoise and the Hare
According to this article by Daniel Tammet, the century in the 1000's with the least primes is the 1300's with 11 primes, and the century with the most primes is the 1400's with 17 primes. However, you would not guess that looking at the first few primes in each century. The 1300's start with a bang. 1301, 1303, and 1307 are all primes, and 1327 is already the 6th prime in the 1300's. The 1400's have a slow start. The first prime in the 1400's is 1409, and it takes until 1423 to get to the second prime. Then, the 1300's fall behind. There is a record prime gap between 1327 and 1361, and another big gap between 1381 and 1399. The 1400's catch up quickly. There are a lot of primes from 1427 to 1499, including the prime quadruplet 1481, 1483, 1487, 1489. It is like the tortoise and the hare!
Here are the primes in the 1300's. 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399 Here are the primes in the 1400's. 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499 
20211212, 15:53  #2 
Jun 2003
Oxford, UK
2^{5}·3^{2}·7 Posts 
The calendar system used in the West is totally arbitrary, based on the incarnation of Christ, as determined in a roundabout way by Dionysius Exiguus. It is not known why he called his then current year "525". His idea became concreted through Bede and the rest "is history".
These days, my childhood AD, has been replaced by CE. How long until CE becomes EF? Last fiddled with by robert44444uk on 20211212 at 15:54 
20211213, 14:49  #3 
Feb 2017
Nowhere
2·2,671 Posts 
One part of the new Prime ktuplets page is the Patterns of prime ktuplets & the HardyLittlewood constants. Assuming the prime ktuplets conjecture is true, the largest number of primes which can occur infinitely often within an interval of length 100, is 23.
I checked the patterns for 21tuplets, and came up with cases for both patterns where the tuplets were between two consecutive multiples of 100. I didn't bother slogging through the patterns for 22 and 23tuples to produce such cases for them. AFAIK computations have not yet produced examples of the cases of 21tuplets I came up with, and the Prime ktuplets page does not indicate any ktuplets for k > 21. The following example gives a 21tuplet of which 20 lie between consecutive multiples of 100. 39433867730216371575457664399 + [0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84] Last fiddled with by Dr Sardonicus on 20211213 at 14:50 Reason: fignix topsy 
20220101, 22:05  #4 
May 2018
3^{5} Posts 
Happy New Year!
Happy New Year! This century has a lot of primes.

20220103, 23:27  #5 
Jan 2022
2_{10} Posts 
I have searched through prime patterns in centuries up to 1e16. In total there are 162002578 possible patterns in centuries, that can repeat. That is assuming my calculations is correct. Counting a century with no primes as one possible pattern, with 1 prime there is 40 possible patterns. With 12 primes in the century the pattern count peaks at 27836859 patterns. If a century have 23 primes, there can be 20 different patterns, 6 of length 96, and 14 of length 98.
The first century that repeat an earlier century is 390500 and 480800, with 5 primes. The last 2 digits of the primes is {3, 27, 39, 53, 81}. i have found repeating pattern for centuries with 16 or less primes. For 17 primes, i expect the search need to reach at least 5e17. There are 108 centuries with 17 primes < 1e16. My guess is i need ~2000 patterns to find the first repeating century with 17 primes. I have only found 3 centuries with 18 primes, the first is 122853771370900. The attached file shows possible pattern counts for each primecount in a century. 
20220104, 00:06  #6  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
9686_{10} Posts 
Quote:


20220104, 09:56  #7 
Romulan Interpreter
"name field"
Jun 2011
Thailand
2684_{16} Posts 
I know a century with no primes.

20220104, 20:02  #8 
Dec 2008
you know...around...
2·353 Posts 
If that's what you like
Who doesn't?
Code:
(20:40) gp > primepi(1671900)primepi(1671800) %1 = 0 10+ years ago I crunched some numbers, but the calculation is on hiatus since. My program was too slow to cross the finish line, maybe someone can fill in the three missing terms for successive record minimal number of primes: Code:
#primes floor(p/1000) 168 0 135 1 127 2 120 3 119 4 114 5 107 7 106 10 103 11 102 14 98 16 94 18 92 29 90 38 88 40 85 43 80 64 76 88 73 168 71 180 69 212 68 293 67 356 63 452 61 555 59 638 58 871 54 913 53 1637 52 2346 46 3279 43 7176 42 14420 38 15369 36 36912 35 51459 34 96733 33 113376 31 141219 28 200315 27 233047 26 729345 25 951847 24 1704275 23 1917281 22 2326985 21 2937877 20 6973534 18 7362853 17 12838437 16 26480476 15 34095574 13 162661473 12 304552694 10 378326417 9 1252542156 8 3475851270 7 6603973861 6 7613200181 5 21185697626 4 81216177240 3 ??? 2 ??? 1 ??? 0 13893290219204 Code:
#primes floor(p/1000) 168 0 135 1 127 2 120 3 119 4 117 6 112 9 109 12 108 15 106 32 102 42 99 67 98 70 97 92 96 136 95 176 94 267 92 450 88 11281 86 40268 79 311773 78 462387 76 3458886 75 4312023 73 12152009 71 18135787 70 166007963 69 1055164750 68 7967879841 67 61681879516 Code:
#primes p 168 1 167 3 166 4 165 6 164 8 163 18 162 30 161 44 160 48 159 74 158 80 157 84 156 114 155 140 154 150 153 168 152 180 151 198 150 200 149 258 148 270 147 272 146 354 145 360 144 390 143 398 142 420 141 422 140 654 139 662 138 692 137 774 136 830 135 858 134 860 133 972 132 1052 131 1110 130 1202 129 1232 128 1308 127 1328 126 1584 125 1608 124 1614 123 1628 122 2144 121 2154 120 2162 119 2442 118 2448 117 3618 116 3632 115 3780 114 3918 113 3924 112 3930 111 4374 110 5882 109 6330 108 6362 107 6368 106 6380 105 6390 104 7592 103 9830 102 9932 101 10338 100 10344 99 10664 98 10668 97 10772 96 13340 95 15804 94 15810 93 18062 92 18132 91 18134 90 18258 89 18314 88 18354 87 18368 86 18372 85 37538 84 37550 83 37580 82 37590 81 37592 80 62234 79 63744 78 63804 77 63810 76 63842 75 63864 74 87642 73 87650 72 87720 71 87768 70 142232 69 142238 68 180348 67 180372 66 180380 65 180548 64 249672 63 287342 62 287348 61 338582 60 359714 59 359720 58 359732 57 359748 56 359762 55 359768 54 637940 53 912980 52 913040 51 913104 50 913184 49 1467360 48 1467444 47 2515922 46 3279000 45 3760578 44 5832714 43 6033932 42 7175654 41 7175658 40 7175678 39 11330162 38 13009824 37 15369200 36 15369204 35 15369210 34 36912020 33 40581774 32 51459114 31 78150732 30 78150818 29 107282508 28 167833710 27 167833712 26 172154108 25 172154114 24 172154132 23 172154138 22 687704484 21 687704492 20 1403621768 19 2140311662 18 2247336164 17 5740961372 16 5740961384 15 7362853034 14 7362853038 13 60120983610 12 88344840308 11 190224606152 10 191218747290 9 499543941588 8 851374997262 7 1745499026868 6 2786121452552 5 ??? 4 ??? 3 ??? 2 ??? 1 ??? 0 1693182318746372 Code:
#primes p 168 2 167 3 166 5 165 11 164 23 163 41 162 53 161 71 160 73 159 101 158 131 157 137 156 139 155 157 154 173 153 191 152 239 151 331 150 337 149 347 148 349 147 353 146 383 145 389 144 641 143 643 142 673 141 727 140 809 139 821 138 881 137 937 136 1427 135 1429 134 1481 133 1483 132 1973 131 1979 130 1987 129 1993 128 3299 127 3301 126 3307 125 3313 124 5381 123 5399 122 5407 121 5413 120 5431 119 6029 118 8513 117 8563 116 8663 115 8951 114 14387 113 14407 112 14699 111 19373 110 19417 109 21313 108 41843 107 41879 106 41887 105 41947 104 56431 103 56437 102 56443 101 266921 100 266947 99 266971 98 267131 97 267139 96 374677 95 449951 94 2209661 93 2209663 92 2209667 91 2209687 90 2372413 89 2372417 88 40268021 87 40268297 86 40268381 85 40268387 84 106291733 83 106291781 82 564911453 81 564911467 80 649964701 79 3583164401 78 3583164413 77 3583164517 76 14982264191 75 24164578853 74 24164578861 73 83653909841 72 5358759792797 71 5358759792817 70 5358759792851 
20220104, 23:49  #9  
Feb 2017
Nowhere
12336_{8} Posts 
Quote:
It seems the maximal k for which there are admissible patterns for prime ktuplets in (x, x + 998) or (x, x + 1000) is k = 163. 

20220105, 00:02  #10  
Jan 2022
2 Posts 
Quote:
Code:
#Primes, first millennia 1: 4911417538051000 2: 1220240682256000 3: 243212983784000 Code:
#Primes, Last known millennia 64: 8643635576221000 65: 3215158032196000 66: 7456069837969000 67: 1176646877107000 I notice that the first list misses some entries, if they don't improve previous entries. The list with all entries: Code:
Missing entries between 163 and 121 are all unknown. #primes: floor(p/1000) 168 0 135 1 127 2 120 3 119 4 118 Unknown 117 6 116 Unknown 115 Unknown 114 5 113 Unknown 112 9 111 Unknown 110 8 109 12 108 15 107 7 106 10 105 13 104 17 103 11 102 14 101 26 100 22 99 36 98 16 97 51 96 39 95 30 94 18 93 55 92 29 91 57 90 38 89 48 88 40 87 61 86 45 85 43 84 66 83 73 82 97 81 69 80 64 79 118 78 202 77 143 76 88 75 175 74 194 73 168 72 256 71 180 70 370 69 212 68 293 67 356 66 515 65 484 64 744 63 452 62 698 61 555 60 690 59 638 58 871 57 1349 56 1089 55 1974 54 913 53 1637 52 2346 51 3965 50 3362 49 3651 48 5105 47 4118 46 3279 45 11355 44 13256 43 7176 42 14420 41 32166 40 20941 39 29248 38 15369 37 43891 36 36912 35 51459 34 96733 33 113376 32 170895 31 141219 30 266116 29 280378 28 200315 27 233047 26 729345 25 951847 24 1704275 23 1917281 22 2326985 21 2937877 20 6973534 19 9274984 18 7362853 17 12838437 16 26480476 15 34095574 14 186020657 13 162661473 12 304552694 11 548261871 10 378326417 9 1252542156 8 3475851270 7 6603973861 6 7613200181 5 21185697626 4 81216177240 3 243212983784 2 1220240682256 1 4911417538051 0 13893290219204 

20220105, 00:04  #11  
If I May
"Chris Halsall"
Sep 2002
Barbados
10,181 Posts 
Quote:
To share, I also enjoy instrumentals. Last fiddled with by chalsall on 20220105 at 00:07 Reason: s/, I enjoy/, I also enjoy/; # Please forgive me my OCD... 

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