This work presents a different approach to twin primes, an approach from the perspective of the Tesla numbers and gives a refresh and new observation of twin primes that could lead us to an answer to the Twin Prime Conjecture problem. We expose a peculiar relation between twin primes and the generation of prime numbers with Tesla numbers. Tesla numbers seem to be present in so many domains like time, vibration and frequency <a href="#ref1">[1]</a>, and the space between twin primes is not the exception. Let us say that twin primes are more than just prime numbers plus 2 or minus 2, and Tesla numbers are more involved with twin primes than we think, and hopefully, this approach give us a better understanding of the distribution of the twin pairs.
In number theory, Twin Prime Conjecture (TPC) is one of the oldest problems, it says that “There exist infinitely many primes p such that p + 2 is a prime” [
To begin, it is known that aside from the number 3, there is no other prime number with its digital root being a Tesla number (3, 6, 9), [
There is no evidence before the nineteenth century about the awareness of twin primes. There is what we could call an approach of them, mentioned by Polignac in one of the two theorems in his paper in 1849 called “New Research on Prime Numbers” [
In 1879, Glaisher published “An Enumeration of Prime-Pair” where he describes a prime pair being two prime numbers separated only by one number.
After counting all the twin pairs in the first and second million he concluded: “there can be little or no doubt that the number of prime pairs is unlimited; but it would be interesting, though probably not easy, to prove this”. Glaisher is seen as the originator of the TPC [
It is due to mention the Fermat’s primality test or Fermat’s little theorem that states that, if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p:
a p − a p = Integer (1)
This test has its exceptions because there are numbers called pseudo primes, that satisfy the primality test but they are not prime numbers, such as 341, that is a base 2 pseudo prime.
It is an ancient method that consists in marking in a table the multiples of every known prime number for example the multiples of 2 are 4, 6, 8, 10, 12, 14, 16, 18, etc. The unmark numbers are primes. In
The numbers 2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 47, 59, 61, 67, 71, 73, 79, 83, 89, 97 are the prime numbers from 0 to 100.
As mentioned the infinity of prime numbers also has repercussion in the distribution
of twin pairs within the infinity where same mathematicians had work in order to achieve the knowledge of how the twin pairs distribute to infinity mathematicians like Brun, Hardy and Littlewood.
Brun’s constant is defined to be the sum of the reciprocals of all twin primes:
B = ( 1 3 + 1 5 ) + ( 1 5 + 1 7 ) + ( 1 11 + 1 13 ) + ⋯ (2)
Recent calculations of the constant gave us:
B = 1.9021605831 ⋯
α = ∏ P > 2 ( 1 − 1 ( P − 1 ) 2 ) ≈ 0.6601618158 ⋯ (3)
Conjecture (Hardy and Littlewood). For every integer k > 0, there are infinitely many prime pairs p, p + 2k, and the number π2k(x) of such pairs less than x is:
π 2 k ~ 2 α ∏ P > 2 P I k P − 1 P − 2 ⋅ l i 2 ( x ) (4)
From the binary series (2n) shown in
1 + 2 = 3, 2 + 4 = 6, 4 + 8 = 12, 8 + 16 = 24, then 3 + 6 = 9, 6 + 12 = 18, 12 + 24 = 36.
From this point and forward the results of the next level are obtained by vertically summing: 12 + 18 = 30, 24 + 36 = 60 and at last 30 + 60 = 90.
The resulting Tesla numbers in
These numbers have Tesla numbers 3, 6 and 9 as their digital root δ(n).
δ ( 12 ) = 1 + 2 = 3 (5)
δ ( 24 ) = 2 + 4 = 6 (6)
δ ( 18 ) = 1 + 8 = 9 (7)
δ ( 30 ) = 3 + 0 = 3 (8)
δ ( 36 ) = 1 + 8 = 9 (9)
δ ( 60 ) = 6 + 0 = 6 (10)
δ ( 90 ) = 9 + 0 = 9 (11)
Curiously enough, the digital root of the prime vector (12) is 9, a Tesla number (13):
11 , + 13 , + 17 , + 19 , + 23 , + 29 , + 31 , + 37 (12)
11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 = 180 , δ ( 180 ) = 9 (13)
To this vector first you add 30, then 60 and continues 90, 120, 150, 180 adding 30 every time. The vector added to the 30n Tesla module:
30 n , + 11 , + 13 , + 17 , + 19 , + 23 , + 29 , + 31 , + 37 (14)
This is the Prime Number Generator Vector (PNGV). By adding 30n to prime numbers the resulting numbers are prime numbers also.
Twin Prime numbers are generated by Tesla numbers.
As we can see in
In each channel, there are 3 columns originated by their progression and the PNGV.
In the first channel, the column in the middle is a Tesla number obtained from its respective progression 6 (3n − 1) when n = 1, we have: 6 (3 × 1 − 1) = 12, in the left column, we have 12 − 1 = 11 that’s a prime number, in the right column we have 12 + 1 = 13 the twin prime number of 11. Then, we apply the PNGV to the left, middle and right columns to obtain potential twin prime numbers as shown in the table. This means we sum 30 to the columns in first row to obtain the second row, then we sum another 30 to the second row to obtain the third one, and we repeat that process arbitrarily. This method is present in every distribution channel and has a very good approximation on generating prime numbers and twin pairs, but has some exceptions.
Let it be noticed that at the middle of twin prime numbers, there is a Tesla number in the 1st, 2nd and 4th channel.
In the third channel (6 (3n + 1)), however, we observe a peculiar situation where even though prime numbers were generated their twin prime numbers were not, due to the numbers of its right column are multiples of five meaning that none of those numbers are prime numbers.
A similar situation occurs on the fifth channel (18 (n + 1)), where its left column is empty due to its generated numbers are again multiples of 5.
These channels allow us to perform a quicker identification of prime numbers.
As we can see in
1st channel: δ [6 (3n − 1)] = Tesla Number.
2nd channel: δ (18n) = Tesla Number.
3rd channel: δ [6 (3n + 1)] = Tesla Number.
4th channel: δ (30n) = Tesla Number.
5th channel: δ (18n + 1) = Tesla Number.
Also, notice that the digital root of the numbers in the right and left columns has been obtained.
From
The numbers in green are prime numbers.
The numbers in pale yellow are composite numbers. In
The numbers in pink are square prime numbers. For example: 72 = 49.
The numbers in blue are square prime numbers times another prime number. For example: 112 = 121.
The numbers in red are cubic prime numbers. For example: 73 = 343.
At the center of every twin pair, there is just one number and that number is a Tesla number, which in some cases generates twin primes as we could see in the channels 1st, 2nd and 4th, and others just prime numbers as in the channels 3rd and 5th.
Thanks to the distribution of the channels in the 3rd and 4th channels, we see in a practical way that complete columns of generated numbers are discarded as primes, given their nature of being multiples of 5.
We can state that “between a twin pair there is always one Tesla number”.
Our eternal gratitude to the engineer Carlos Enrique Ramirez Escamilla for all his institutional support to carry out this research, and Dr. Larisa Burtseva for her guidance aimed at carrying out this research.
The authors declare no conflicts of interest regarding the publication of this paper.
Mendoza-Navarrete, J., Perez, H., Ojeda, I., Verdugo, V., Sandoval, G.L., Garcia, R.J. and Alvarez, J. (2024) Distribution and Relation of Primes with Tesla Numbers 3, 6, and 9. Journal of Applied Mathematics and Physics, 12, 1021-1027. https://doi.org/10.4236/jamp.2024.124062