chevron_left
Back to "Mathematics > Prime Sets II"
Continue to "Music Creation > read_me_first.txt"
chevron_right12-12-2003
Prime twins
Prime twins, those mysterious prime number partners, quickly became a popular branch of mathematics thousands of years ago. 11 and 13 are twins, 17 and 19 as well, just like 29 and 31, etc.. - in other words: two prime numbers with a difference of 2. Unfortunately, until today only very few useful facts about prime twins are known - they seem to occur only sporadically and without any pattern.
Reason enough for me to spend some time to possibly discover something. No sooner said than done - during the night from 11/24/2003 to 11/25/2003 I discovered an interesting regularity among the prime twins. Unfortunately, I could follow this regularity only to some extent due to inadequate software, which is why I am very grateful for any help that has been given and possibly will be given in the future.
1.1.a.) among the first 10 numbers there are 4 prime numbers
1.1.b.) among the first 10 primes there are 4 prime twins
2.1.a.) among the first 100 numbers there are 25 primes
2.1.b.) among the first 100 primes there are 25 prime twins
3.1.a.) among the first 1000 numbers there are 168 primes
3.1.b.) among the first 1000 primes there are 174 prime twins
4.1.a.) among the first 10000 numbers there are 1229 primes
4.1.b.) among the first 10000 primes there are 1270 prime twins
Danke an Henrik Schulz für 5.1:
5.1.a.) among the first 100000 numbers there are 9592 primes
5.1.b.) among the first 100000 primes there are 10250 prime twins
Danke an Eugen Bauhof für 6.1 - 7.3:
6.1.a.) among the first 1000000 numbers there are 78498 primes
6.1.b.) among the first 1000000 primes there are 86027 prime twins
6.2.a.) among the first 2000000 numbers there are 148933 primes
6.2.b.) among the first 2000000 primes there are 163766 prime twins
6.3.a.) among the first 3000000 numbers there are 216816 primes
6.3.b.) among the first 3000000 primes there are 239003 prime twins
6.4.a.) among the first 4000000 numbers there are 283146 primes
6.4.b.) among the first 4000000 primes there are 312703 prime twins
6.5.a.) among the first 5000000 numbers there are 348513 primes
6.5.b.) among the first 5000000 primes there are 385375 prime twins
6.6.a.) among the first 6000000 numbers there are 412849 primes
6.6.b.) among the first 6000000 primes there are 457399 prime twins
6.7.a.) among the first 7000000 numbers there are 476648 primes
6.7.b.) among the first 7000000 primes there are 528528 prime twins
6.8.a.) among the first 8000000 numbers there are 539777 primes
6.8.b.) among the first 8000000 primes there are 599296 prime twins
6.9.a.) among the first 9000000 numbers there are 602489 primes
6.9.b.) among the first 9000000 primes there are 669294 prime twins
7.1.a.) among the first 10000000 numbers there are 664579 primes
7.1.b.) among the first 10000000 primes there are 738597 prime twins
7.2.a.) among the first 11000000 numbers there are 726517 primes
7.2.b.) among the first 11000000 primes there are 807787 prime twins
7.3.a.) among the first 12000000 numbers there are 788060 primes
7.3.b.) among the first 12000000 primes there are 876670 prime twins
So the density of the prime twins within the primes seems to be closely related to the density of the primes in the integers (1, 2, 3, etc.). Now these are very basic, number-theoretic facts - so, I thought, I'll follow this up in an equally basic, number-theoretic way. The very first approximation formula that ever existed that could be used to estimate prime numbers is x/ln(x). This tells approximately how many prime numbers lie between 1 and x. Conversely, the formula x*ln(x) gives approximately the xth prime number. Since x*ln(x) is to be understood as the inverse of x/ln(x), one can probably use this the other way round: with this, one should be able to estimate how many numbers are necessary to contain x prime twins. To accomplish this, however, something more is needed, as can be seen in the following table:
x integers contain --> | y primes contain --> | z prime twins |
x | y = x / ln(x) | z = [x / ln(x)] / ln[x / ln(x)] |
x = y * ln(x) | y | z = y / ln(y) |
x = z * ln(y) * ln(x) | y = z * ln(y) | z |
The pink colored formulas in the lower left corner of the table are unfortunately not solvable this way, if you don't want to look for the needle in the haystack. Let's take y = z * ln(y) as an example. To get y as a solution, we would have to know y already - since this is a contradiction, it is not that easy. Therefore, we have to somehow try to calculate or at least estimate y in a roundabout way. My approach now is to estimate ln(y) using ln(z). How big is the difference between both numbers and how could we compensate it?
Here are my thoughts:
1a.) 100 / ln(100) = 21,7147
1b.) 21,7147 * ln(21,7147) = 66,8377
1c.) 21,7147 * ln(66,8377 * √2) = 98,7768
2a.) 1000 / ln(1000) = 144,7648
2b.) 144,7648 * ln(144,7648) = 720,2210
2c.) 144,7648 * ln(720,2210 * √2) = 1002,6603
3a.) 10000 / ln(10000) = 1085,7362
3b.) 1085,7362 * ln(1085,7362) = 7589,3108
3c.) 1085,7362 * ln(7589,3108 * √2) = 10076,7933
4a.) 100000 / ln(100000) = 8685,8896
4b.) 8685,8896 * ln(8685,8896) = 78776,2861
4c.) 8685,8896 * ln(78776,2861 * √2) = 100938,21
In my opinion, this is accurate enough for now - so we can dare to use the following approximation formula:
a.) x / ln(x) = n
b.) n * ln(n) = m
c.) n * ln(m * √2) ~ x
d.) n * ln(n * ln(n) * √2) ~ x
The root of 2 is only an approximation factor - otherwise you could nest this approximation more and more and make it more accurate. It would then look like this: x = n * ln(n * ln(n * ln(n * ln(n * ...))))
However, since this is fundamentally only about estimations, it should be kept simple. Now we can estimate the x from x/ln(x) and thus modify the above table a bit to make it more useful for our purposes. As you can see in the following table, despite the strong simplification, it is still rather extreme to deduce x from z.
x integers contain --> | y primes contain --> | z prime twins |
x | y = x / ln(x) | z = [x / ln(x)] / ln[x / ln(x)] |
x = y * ln(y * ln(y) * √2) | y | z = y / ln(y) |
x = z * ln(y) * ln(y * ln(y) * √2) | y = z * ln(z * ln(z) * √2) | z |
x = z * ln(z * ln(z * ln(z) * √2)) * ln(z * ln(z * ln(z) * √2)) * ln(z * ln(z * ln(z) * √2) * √2) |
If one were to write out the formula for x in the third row of this table completely, the pink colored formula would emerge. If we wanted to simplify this, we would have to make the whole approximation less precise. For this reason it is advisable to first calculate
y = z * ln(z * ln(z) * √2)
and then transfer y and z into the formula
x = z * ln(y) * ln(y * ln(y) * √2)
Here is a table by Eugen Bauhof, which shows that the factor connecting the prime set with the twin set apparently converges, i.e. it takes on a specific value at infinity. Similarly as x/ln(x) approximates the number of primes among the integers, n * x/ln(x) would approximate the number of prime twins among the primes. I'm curious to know exactly how big this factor turns out to be! Here is the table: