Distribution and Relation of Primes with Tesla Numbers 3, 6, and 9

Javier Mendoza-Navarrete¹, Huetzin Perez², Ismael Ojeda³, Victor Verdugo³, Gabriel Luna Sandoval⁴, Ricardo Jimenez Garcia⁵, Jaime Alvarez⁶

¹Departametno de Matemáticas, CBTis 256 DGETI, Cabo San Lucas, México.
²DCI Departamento de Fisica Médica, Universidad de Guanajuato, Leon, México.
³División de Ingenieria en Sistemas Computacionales, Tecnológico Nacional de México ITES Los Cabos, Cabo San Lucas, México.
⁴Departamento de Ingenieria Industrial en Manufactura, Universidad Estatal de Sonora, Navojoa, México.
⁵Departamento de Eléctrica-Electrónica, Tecnológico Nacional de México Campus Mexicali, Mexicali, México.
⁶Guanajuato, México.

DOI: 10.4236/jamp.2024.124062                

Abstract

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 [1], 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.

Keywords

Tesla Numbers, Prime Number, Twin Prime Numbers, Twin Pair

1. Introduction

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” [2] [3] . In other words, twin primes are prime numbers that differ by two, such as 11 and 13, or 41 and 43 and many others [2] . Do not confuse them with consecutive prime numbers for they are numbers that cannot only be close together, but also be far apart [3] . However, it could be said to step towards the famous twin prime conjecture that there are infinitely many prime pairs p and p + 2, the gap here being 2 [4] . With that observation established, we present a different perspective that establishes a close relation and interaction with Tesla numbers.

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), [1] that sentence provides a simple beginning to identify prime numbers, we could say that provides a rough method to identify prime numbers that involve Tesla numbers.

2. Historical Context

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” [2] [5] . The theorem says: “Every even number is equal to the difference of two consecutive prime numbers in an infinitude of ways”. When changing “every even number” by the number two, there is a reference to the twin primes and their infinitude [2] . It is quite a reach to consider this the origin of the TPC.

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 [2] .

2.1. Prime Numbers Identification Methods

2.1.1. Fermat’s Primality Test

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 a^p − a is an integer multiple of p:

$\frac{a^p-a}{p}=\text{Integer}$ (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.

2.1.2. Sieve of Eratosthenes

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 Figure 1, this method is shown.

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.

2.2. Twin Primes’ Distribution

As mentioned the infinity of prime numbers also has repercussion in the distribution

Figure 1. Sieve of Eratosthenes from 0 to 100.

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.

2.2.1. Brun’s Constant

Brun’s constant is defined to be the sum of the reciprocals of all twin primes:

$B=\left(\frac{1}{3}+\frac{1}{5}\right)+\left(\frac{1}{5}+\frac{1}{7}\right)+\left(\frac{1}{11}+\frac{1}{13}\right)+\cdots$ (2)

Recent calculations of the constant gave us:

$B=1.9021605831\cdots$

2.2.2. Hardy and Littlewood Twin Prime Constant

$\alpha =\underset{P>2}{{\displaystyle \prod }}\left(1-\frac{1}{{\left(P-1\right)}^2}\right)\approx 0.6601618158\cdots$ (3)

Conjecture (Hardy and Littlewood). For every integer k > 0, there are infinitely many prime pairs p, p + 2k, and the number π_2k(x) of such pairs less than x is:

${\pi }_{2k}~2\alpha {\displaystyle \underset{\begin{array}{c}P>2\\ PIk\end{array}}{\prod }\frac{P-1}{P-2}\cdot l{i}_2\left(x\right)}$ (4)

3. Methodology

From the binary series (2n) shown in Figure 2, Tesla numbers are obtained by horizontally summing the powers of two on the first level giving us the result to the next level as follows:

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 Figure 3 are prime number generators.

These numbers have Tesla numbers 3, 6 and 9 as their digital root δ(n).

Figure 2. Tesla numbers from powers of 2.

Figure 3. Prime numbers generators.

$\delta \left(12\right)=1+2=3$ (5)

$\delta \left(24\right)=2+4=6$ (6)

$\delta \left(18\right)=1+8=9$ (7)

$\delta \left(30\right)=3+0=3$ (8)

$\delta \left(36\right)=1+8=9$ (9)

$\delta \left(60\right)=6+0=6$ (10)

$\delta \left(90\right)=9+0=9$ (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)

$11+13+17+19+23+29+31+37=180,\delta \left(180\right)=9$ (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:

$30n,+11,+13,+17,+19,+23,+29,+31,+37$ (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 Figure 4, there are five distribution channels each with a different progression.

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 1^st, 2^nd and 4^th 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 Figure 5, every channel the column in the middle is a Tesla Number:

1^st channel: δ [6 (3n − 1)] = Tesla Number.

2^nd channel: δ (18n) = Tesla Number.

3^rd channel: δ [6 (3n + 1)] = Tesla Number.

4^th channel: δ (30n) = Tesla Number.

5^th channel: δ (18n + 1) = Tesla Number.

Also, notice that the digital root of the numbers in the right and left columns has been obtained.

Figure 4. The five distribution channel table.

From Figure 5, we could say that “A prime number is an odd number that comes before or after a Tesla number. Also, it is divisible by 1 and by itself” and “Twin Prime numbers are odd numbers that come before and after a Tesla number, also divisible by 1 and themselves”, that is why the numbers 2 and 3 are not twin prime numbers.

The numbers in green are prime numbers.

Figure 5. Digital roots of the five distribution channels.

Figure 6. Composite numbers distribution.

Figure 7. Square and cubic composite numbers distribution.

The numbers in pale yellow are composite numbers. In Figure 6 and Figure 7, we see the distribution of composite numbers.

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.

4. Conclusions

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 1^st, 2^nd and 4^th, and others just prime numbers as in the channels 3^rd and 5^th.

Thanks to the distribution of the channels in the 3^rd and 4^th 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”.

Acknowledgements

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.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

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[3]	Nazardonyavi, S. (2012) Some History about Twin Prime Conjecture. arXiv: 1205.0774. https://doi.org/10.48550/arXiv.1205.0774
[4]	Soundararajan, K. (2007) Small Gaps between Prime Numbers: The Work of Goldston-Pintz-Yildirim. Bulletin of the American Mathematical Society, 44, 1-18. https://doi.org/10.1090/S0273-0979-06-01142-6
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