mersenneforum.org Primes in centuries: The Tortoise and the Hare
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 2021-12-12, 03:07 #1 Bobby Jacobs     May 2018 35 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 25·32·7 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 These days, my childhood AD, has been replaced by CE.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 2021-12-12 at 15:54
 2021-12-13, 14:49 #3 Dr Sardonicus     Feb 2017 Nowhere 2·2,671 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 35 Posts Happy New Year! Happy New Year! This century has a lot of primes.
2022-01-03, 23:27   #5
kog67

Jan 2022

210 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 prime-count in a century.
Attached Files
 PatternCountCentury.txt (287 Bytes, 19 views)

2022-01-04, 00:06   #6
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

968610 Posts

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.

 2022-01-04, 09:56 #7 LaurV Romulan Interpreter     "name field" Jun 2011 Thailand 268416 Posts I know a century with no primes.
2022-01-04, 20:02   #8
mart_r

Dec 2008
you know...around...

2·353 Posts
If that's what you like

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

2022-01-04, 23:49   #9
Dr Sardonicus

Feb 2017
Nowhere

123368 Posts

Quote:
 Originally Posted by mart_r Who doesn't? Code: (20:40) gp > primepi(1671900)-primepi(1671800) %1 = 0
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.

2022-01-05, 00:02   #10
kog67

Jan 2022

2 Posts

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

2022-01-05, 00:04   #11
chalsall
If I May

"Chris Halsall"
Sep 2002

10,181 Posts

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.

Last fiddled with by chalsall on 2022-01-05 at 00:07 Reason: s/, I enjoy/, I also enjoy/; # Please forgive me my OCD...

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