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A pattern for counting twin
primes may be detected as follows (counting from 4 onwards):
|
Range: 4 - |
Approximation |
#twin primes |
#primes |
ratio(approx.) |
Decimal value |
|
7 |
|
1 |
2 |
1:2 |
0.5 |
|
100 |
100x4^0 (100) |
7 |
23 |
1:3 |
0.30434 |
|
500 |
100x4^1 (400) |
23 |
93 |
1:4 |
0.24731 |
|
2,000 |
100x4^2 (1,600) |
60 |
301 |
1:5 |
0.19933 |
|
7,000 |
100x4^3 (6,400) |
161 |
898 |
1:6 |
0.17928 |
|
20,000 |
100x4^4 (25,600) |
341 |
2,260 |
1:7 |
0.15088 |
|
90,000 |
100x4^5 (102,400) |
1,115 |
8,711 |
1:8 |
0.12799 |
|
500,000 |
100x4^6 (409,600) |
4,564 |
41,536 |
1:9 |
0.10988 |
|
1,800,000 |
100x4^7 (1,638,400) |
13,555 |
135,062 |
1:10 |
0.10036 |
This shows the approximate
ratio #twin primes: #primes up to 100(4^n) is 1: (n+3)
Notation: Let P(n)
represent #primes up to n (approx.) and TP(n) represent #twin primes up to n
[counting from 4]
By observation: TP(100(4^n)):P(100x4^n) = 1:(n+3)
That is: TP(100(4^n))
= P((100(4^n))/(n+3)
------: __________ =
100(4^n)/[ (n+3)(Log(100(4^n)) - 1) ]
[Using n/(Log n -1) = P(n)]
------:___________ =
100(4^n)/[ (n+3)(nLog4 + Log100 -1) ]
Let 100(4^n) = x. Then n =
(Log x - Log 100)/Log 4
Therefore: TP(x) = x/((Log x - Log 100)/Log 4 + 3)(Log x -Log 100 + Log 100 -1)
---------: _____ = x Log 4/((Log x -1)(Log x + Log(0.64)))
This gives an estimate
(based on observation) of the number of twin primes.
Other estimates available
are (1): TP(x) = x Log 3.744/(Log x)(Log x) [Hardy and
Littlewood]
A table of comparisons
|
x |
actual #twin primes |
source |
my estimated #twin primes |
other estimates(1) |
|
|
7 |
2 |
count |
6.84 |
2.44 |
|
|
100 |
8 |
count |
9.24 |
6.22 |
|
|
500 |
24 |
own program |
23.04 |
17.09 |
|
|
2,000 |
61 |
" |
58.70 |
45.70 |
|
|
7,000 |
162 |
" |
146.96 |
117.88 |
|
|
20,000 |
342 |
" |
329.27 |
269.20 |
|
|
90,000 |
1,116 |
" |
1,093.67 |
913.02 |
|
|
500,000 |
4,565 |
" |
4,510.79 |
3,833.28 |
|
|
1,800,000 |
13,556 |
" |
13,339.01 |
11,454.42 |
|
|
4,000,000 |
26,833 |
" |
26,461.6 |
22,850.43 |
|
|
10,000,000 |
58,980 |
" |
58,511.2 |
50,815.63 |
|
|
100,000,000 |
440,312 |
" |
442,727.07 |
389,057.17 |
|
|
10^9 |
3,424,506 |
3,466,357.75 |
3,074,031.99 |
|
|
|
10^10 |
27,412,679 |
" |
27,874,508.14 |
24,899,659.16 |
|
|
10^11 |
224,376,048 |
" |
229,009,438 |
205,782,307.1 |
|
|
10^14 |
135,780,321,665 |
" |
139,607,311,5xx |
127,039,077,4xx |
|
|
10^15 |
1,177,209,242,302 |
1,212,410,056,xxx |
1,106,651,518,xxx |
Added 2011 |
|
|
10^16 |
10,304,195,697,298 |
“ |
10,627,432,29x,xxx |
9,726,429,36x,xxx |
|
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