Primes in centuries: The Tortoise and the Hare

Bobby Jacobs · 2021-12-12, 03:07

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

robert44444uk · 2021-12-12, 15:53

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These days, my childhood AD, has been replaced by CE.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?

Dr Sardonicus · 2021-12-13, 14:49

One part of the new Prime k-tuplets page is the Patterns of prime k-tuplets & the Hardy-Littlewood constants. Assuming the prime k-tuplets 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 21-tuplets, 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 23-tuples to produce such cases for them.

AFAIK computations have not yet produced examples of the cases of 21-tuplets I came up with, and the Prime k-tuplets page does not indicate any k-tuplets for k > 21. The following example gives a 21-tuplet 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]

Bobby Jacobs · 2022-01-01, 22:05

Happy New Year! This century has a lot of primes.

kog67 · 2022-01-03, 23:27

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 prime-count in a century.

Batalov · 2022-01-04, 00:06

Quote:

Originally Posted by kog67

I have searched through prime patterns in centuries up to 1e16.
I have only found 3 centuries with 18 primes, the first is 122853771370900.

This appears to be a known fact, but still - a good find.

LaurV · 2022-01-04, 09:56

I know a century with no primes.

mart_r · 2022-01-04, 20:02

Quote:

Originally Posted by LaurV

I know a century with no primes.

Who doesn't?

Code:

(20:40) gp > primepi(1671900)-primepi(1671800)
%1 = 0

While we're at it, what about primes in millennia?
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

I've additionally kept track of the last known appearance of #primes in millennia, for p < 10^14:

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

Also, starting from p, there are successively less primes in the interval p+[0..999] (five terms missing):

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

Aaand the last known appearance of #primes in p+[0..999], checked for p < 7*10^12:

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

Dr Sardonicus · 2022-01-04, 23:49

Quote:

Originally Posted by mart_r

Who doesn't?

Code:

(20:40) gp > primepi(1671900)-primepi(1671800)
%1 = 0

<snip>

Nit-pick: in the Gregorian calendar, centuries begin at the start of year 100*k + 1, not 100*k.

It seems the maximal k for which there are admissible patterns for prime k-tuplets in (x, x + 998) or (x, x + 1000) is k = 163.

kog67 · 2022-01-05, 00:02

Quote:

Originally Posted by mart_r

Who doesn't?

While we're at it, what about primes in millennia?
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:

I have found the 3 missing terms :

Code:

#Primes, first millennia
1: 4911417538051000
2: 1220240682256000
3:  243212983784000

Also, i have a few terms for the last known millennia

Code:

#Primes, Last known millennia
64: 8643635576221000
65: 3215158032196000
66: 7456069837969000
67: 1176646877107000

I can't help with the two last lists.

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

chalsall · 2022-01-05, 00:04

Quote:

Originally Posted by Dr Sardonicus

Nit-pick: in the Gregorian calendar, centuries begin at the start of year 100*k + 1, not 100*k.

Yeah. The pedant amoungst us notice things like that. I enjoy their company...

To share, I also enjoy instrumentals.

2021-12-12, 03:07	#1
Bobby Jacobs May 2018 102₁₆ 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

2021-12-12, 15:53	#2
robert44444uk Jun 2003 Oxford, UK 2039₁₀ 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 2021-12-12 at 15:54

2021-12-13, 14:49	#3
Dr Sardonicus Feb 2017 Nowhere 2·2,887 Posts	One part of the new Prime k-tuplets page is the Patterns of prime k-tuplets & the Hardy-Littlewood constants. Assuming the prime k-tuplets 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 21-tuplets, 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 23-tuples to produce such cases for them. AFAIK computations have not yet produced examples of the cases of 21-tuplets I came up with, and the Prime k-tuplets page does not indicate any k-tuplets for k > 21. The following example gives a 21-tuplet 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 2021-12-13 at 14:50 Reason: fignix topsy

2022-01-01, 22:05	#4
Bobby Jacobs May 2018 402₈ Posts	Happy New Year! Happy New Year! This century has a lot of primes.

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2022-01-04, 09:56	#7
LaurV Romulan Interpreter "name field" Jun 2011 Thailand 2×17×293 Posts	I know a century with no primes.