素数表逐级扩大的编制方法

yangchuanju

目前，专业分解网站对于300位以下的整数基本上都分解了，300位以上大整数相当多无法分解；没有完整的素数资源就编制不出完整的素数表。

当前在网上可找到的素数资源库仅21亿多（10位数），吴代业希望扩大到22亿估计有可能，再大不是非专业人员所能做到的。
要让专业人员编制100位素数表，有可能，再大他们也难以办到。

100万位——那是天文数字！

重生888@

100万位——那是天文数字！

100万位，不是无穷大，应该能做到，只是时间问题。一亿亿（18位）以内，应轻易做到！

yangchuanju

1亿亿素数表的编制
1亿亿，10的16次方，
1亿亿内共有279238341033925个素数（279万亿多个素数）；
1亿亿内的最大素数是9999999999999937，比1亿亿小63。
1亿亿的平方根是1亿，内有5761455个素数（576万多个素数）；
1亿内的最大素数是99999989，比1亿小11。

比1亿亿稍大一些的素数阶乘是43#=13082761331670030（1.3*10^16）；
43#内有362597750396740个素数，最大素数是13082761331669941，比43#小89；
43#的平方根是114379899，内有6539376个素数，最大的是114379879，比平方根小20。

要编制1亿亿内的素数表，可用素数阶乘的逐级扩大法计算到43#。

yangchuanju

当前通用的素数表，实际上（可能是）通过2的次方数逐级扩大法编制的，
由2^10=1024扩大到2的20次方（1百万），再逐级扩大到2的31次方（21.47亿）编制出来的。

该表属于16位数字系统，除去一个符号位，最大有效位数是15位；
当前的数字系统已升级到32位，除去符号位，最大有效位数的31位，对应于2^63=9.22*10^18（922亿亿），
数字系统升级以后，不要说编制1亿亿素数表成为可能，编制出900亿亿以内素数表也变得可行；
但要编制1000亿亿素数表，还需等待数字系统进一步升级至64位以后再做定论。

yangchuanju

A006880-29
Number of primes < 10^n.
小于10^n以内的素数个数
1 4
2 25
3 168
4 1229
5 9592
6 78498
7 664579
8 5761455
9 50847534
10 455052511
11 4118054813
12 37607912018
13 346065536839
14 3204941750802
15 29844570422669
16 279238341033925
17 2623557157654233
18 24739954287740860
19 234057667276344607
20 2220819602560918840
21 21127269486018731928
22 201467286689315906290
23 1925320391606803968923
24 18435599767349200867866
25 176846309399143769411680
26 1699246750872437141327603
27 16352460426841680446427399
28 157589269275973410412739598
29 1520698109714272166094258063

yangchuanju

A003618
Largest n-digit prime.
最大的n位素数
1 7
2 97
3 997
4 9973
5 99991
6 999983
7 9999991
8 99999989
9 999999937
10 9999999967
11 99999999977
12 999999999989
13 9999999999971
14 99999999999973
15 999999999999989
16 9999999999999937
17 99999999999999997
18 999999999999999989
19 9999999999999999961
20 99999999999999999989
21 999999999999999999899
22 9999999999999999999973
23 99999999999999999999977
24 999999999999999999999743
25 9999999999999999999999877
26 99999999999999999999999859
27 999999999999999999999999901
28 9999999999999999999999999791
29 99999999999999999999999999973
30 999999999999999999999999999989
31 9999999999999999999999999999973
32 99999999999999999999999999999979
33 999999999999999999999999999999991
34 9999999999999999999999999999999589
35 99999999999999999999999999999999977
36 999999999999999999999999999999999841
37 9999999999999999999999999999999999919
38 99999999999999999999999999999999999941
39 999999999999999999999999999999999999943
40 9999999999999999999999999999999999999983
41 99999999999999999999999999999999999999881
42 999999999999999999999999999999999999999917
43 9999999999999999999999999999999999999999919
44 99999999999999999999999999999999999999999947
45 999999999999999999999999999999999999999999991
46 9999999999999999999999999999999999999999999967
47 99999999999999999999999999999999999999999999959
48 999999999999999999999999999999999999999999999967
49 9999999999999999999999999999999999999999999999943
50 99999999999999999999999999999999999999999999999943
51 999999999999999999999999999999999999999999999999677
52 9999999999999999999999999999999999999999999999999769
53 99999999999999999999999999999999999999999999999999823
54 999999999999999999999999999999999999999999999999999709
55 9999999999999999999999999999999999999999999999999999889
56 99999999999999999999999999999999999999999999999999999407
57 999999999999999999999999999999999999999999999999999999907
58 9999999999999999999999999999999999999999999999999999999851
59 99999999999999999999999999999999999999999999999999999999859
60 999999999999999999999999999999999999999999999999999999999839
61 9999999999999999999999999999999999999999999999999999999999961
62 99999999999999999999999999999999999999999999999999999999999917
63 999999999999999999999999999999999999999999999999999999999999877
64 9999999999999999999999999999999999999999999999999999999999999949
65 99999999999999999999999999999999999999999999999999999999999999731
66 999999999999999999999999999999999999999999999999999999999999999379
67 9999999999999999999999999999999999999999999999999999999999999999889
68 99999999999999999999999999999999999999999999999999999999999999999607
69 999999999999999999999999999999999999999999999999999999999999999999811
70 9999999999999999999999999999999999999999999999999999999999999999999907
71 99999999999999999999999999999999999999999999999999999999999999999999947
72 999999999999999999999999999999999999999999999999999999999999999999999883
73 9999999999999999999999999999999999999999999999999999999999999999999999851
74 99999999999999999999999999999999999999999999999999999999999999999999999953
75 999999999999999999999999999999999999999999999999999999999999999999999999809
76 9999999999999999999999999999999999999999999999999999999999999999999999999637
77 99999999999999999999999999999999999999999999999999999999999999999999999999457
78 999999999999999999999999999999999999999999999999999999999999999999999999999937
79 9999999999999999999999999999999999999999999999999999999999999999999999999999749
80 99999999999999999999999999999999999999999999999999999999999999999999999999999989
81 999999999999999999999999999999999999999999999999999999999999999999999999999999827
82 9999999999999999999999999999999999999999999999999999999999999999999999999999999637
83 99999999999999999999999999999999999999999999999999999999999999999999999999999999947
84 999999999999999999999999999999999999999999999999999999999999999999999999999999999601
85 9999999999999999999999999999999999999999999999999999999999999999999999999999999999973
86 99999999999999999999999999999999999999999999999999999999999999999999999999999999999983
87 999999999999999999999999999999999999999999999999999999999999999999999999999999999999727
88 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999959
89 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999779
90 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999857
91 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999919
92 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999217
93 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999827
94 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999433
95 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999947
96 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999731
97 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999841
98 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999929
99 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999379
100 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999203
……

yangchuanju

A132153
Largest prime <= square root of 10^n.
10的n次方平方根以内的最大素数
1 3
2 7
3 31
4 97
5 313
6 997
7 3137
8 9973
9 31607
10 99991
11 316223
12 999983
13 3162277
14 9999991
15 31622743
16 99999989
17 316227731
18 999999937
19 3162277633
20 9999999967
21 31622776589
22 99999999977
23 316227766003
24 999999999989
25 3162277660153
26 9999999999971
27 31622776601657
28 99999999999973
29 316227766016779
30 999999999999989
31 3162277660168279
32 9999999999999937
33 31622776601683777
34 99999999999999997
35 316227766016837933
36 999999999999999989
37 3162277660168379281
38 9999999999999999961
39 31622776601683793293
40 99999999999999999989
41 316227766016837933179
42 999999999999999999899
43 3162277660168379331989
44 9999999999999999999973
45 31622776601683793319839
46 99999999999999999999977
47 316227766016837933199881
48 999999999999999999999743
49 3162277660168379331998873
50 9999999999999999999999877
51 31622776601683793319988927
52 99999999999999999999999859
53 316227766016837933199889291
54 999999999999999999999999901
55 3162277660168379331998893501
56 9999999999999999999999999791
57 31622776601683793319988935437
58 99999999999999999999999999973
59 316227766016837933199889354357
60 999999999999999999999999999989
61 3162277660168379331998893544423
62 9999999999999999999999999999973
63 31622776601683793319988935444299
64 99999999999999999999999999999979
65 316227766016837933199889354443271
66 999999999999999999999999999999991
67 3162277660168379331998893544432689
68 9999999999999999999999999999999589
69 31622776601683793319988935444327133
70 99999999999999999999999999999999977
71 316227766016837933199889354443271843
72 999999999999999999999999999999999841
73 3162277660168379331998893544432718523
74 9999999999999999999999999999999999919
75 31622776601683793319988935444327185329
76 99999999999999999999999999999999999941
77 316227766016837933199889354443271853287
78 999999999999999999999999999999999999943
79 3162277660168379331998893544432718533629
80 9999999999999999999999999999999999999983
81 31622776601683793319988935444327185337143
82 99999999999999999999999999999999999999881
83 316227766016837933199889354443271853371921
84 999999999999999999999999999999999999999917
85 3162277660168379331998893544432718533719547
86 9999999999999999999999999999999999999999919
87 31622776601683793319988935444327185337195547
88 99999999999999999999999999999999999999999947
89 316227766016837933199889354443271853371955491
90 999999999999999999999999999999999999999999991
91 3162277660168379331998893544432718533719554829
92 9999999999999999999999999999999999999999999967
93 31622776601683793319988935444327185337195551381
94 99999999999999999999999999999999999999999999959
95 316227766016837933199889354443271853371955513857
96 999999999999999999999999999999999999999999999967
97 3162277660168379331998893544432718533719555139317
98 9999999999999999999999999999999999999999999999943
99 31622776601683793319988935444327185337195551393147
100 99999999999999999999999999999999999999999999999943
……

yangchuanju

A122121-52
Number of primes <= 10^(n/2).
10的n/2次方以内的素数个数（至26次方）
1 2
2 4
3 11
4 25
5 65
6 168
7 446
8 1229
9 3401
10 9592
11 27293
12 78498
13 227647
14 664579
15 1951957
16 5761455
17 17082666
18 50847534
19 151876932
20 455052511
21 1367199811
22 4118054813
23 12431880460
24 37607912018
25 113983535775
26 346065536839
27 1052370166553
28 3204941750802
29 9773865306521
30 29844570422669
31 91238789797384
32 279238341033925
33 855502559228365
34 2623557157654233
35 8052994747583677
36 24739954287740860
37 76066570954337300
38 234057667276344607
39 720722641159301040
40 2220819602560918840
41 6847673381013822597
42 21127269486018731928
43 65223071241820793398
44 201467286689315906290
45 622647095301172021671
46 1925320391606803968923
47 5956317545928249075039
48 18435599767349200867866
49 57086403558149290301868
50 176846309399143769411680
51 548074549053620897173483
52 1699246750872437141327603