The Periodic Table of Primes

Han-Lin Li¹, Shu-Cherng Fang², Way Kuo^3*

¹Department of Computer Science, City University of Hong Kong, Hong Kong, China.
²Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, USA.
³Hong Kong Institute for Advanced Study, City University of Hong Kong, Hong Kong, China.

DOI: 10.4236/apm.2024.145023                

Abstract

Over millennia, nobody has been able to predict where prime numbers sprout or how they spread. This study establishes the Periodic Table of Primes (PTP) using four prime numbers 2, 3, 5, and 7. We identify 48 integers out of a period $2\times 3\times 5\times 7=210$ to be the roots of all primes as well as composites without factors of 2, 3, 5, and 7. Each prime, twin primes, or composite without factors of 2, 3, 5, and 7 is an offspring of the 48 integers uniquely allocated on the PTP. Three major establishments made in the article are the Formula of Primes, the Periodic Table of Primes, and the Counting Functions of Primes and Twin Primes.

Keywords

Primes, Composites, the Periodic Table of Primes

1. Introduction

One of the greatest theorems of mathematics states that a composite integer can be represented uniquely as a product of primes. Today primes as roots of integers are studied and applied widely to data science, cryptography [1] , systems reliability design [2] , etc. For a long time, many said these integer roots grew like weeds among natural numbers, and nobody could predict where the next primes may sprout [3] . Many believe that primes are unpredictable. Oliver and Soundaranjan [4] investigated the distribution of consecutive primes, while Luque and Lacasa [5] reported patterns of the last digit of primes. Wang [6] adopted pictures to search for regularities of some of the primes, but with significant bias.

Owing to the lack of insight, there is no effective computable formula for counting functions of primes and twin primes. Studies published to predict primality and pattern of primes are ad hoc with tremendous limitations and uncertainties.

Some [5] observed that primes located near each other tend to avoid repeating their last digits, which indicates that primes are not distributed as randomly as theorists often assume. Others such as Tóth [7] found the existence of the primes in k-tuple where the difference of two neighboring primes stays the same. However, few are able to explain what and where these tuple primes occur, nor can they predict the longest tuple primes within an interval.

Initiated by Gauss, there have been studies on counting the total number of primes and twin primes within an interval, but only approximations were obtained [8] [9] . Since primes are the roots of integers, can we identify the roots of primes? Can we find rules for generating composites; and if possible, may we say that, in some aspects, primes are predictable?

Li et al. [10] used 2, 3 and 5 in building a universal color system C₂₃₅ to unify RGB (a light color frame) and CMYK (a pigment color frame). C₂₃₅ represents colors R(red), G(green), and B(blue) by primes 2, 3, and 5, respectively. Consequently, C(cyan), M(magenta), Y(yellow), and K(key black) are represented by $3\times 5=15$ , $2\times 5=10$ , $2\times 3=6$ , and $2\times 3\times 5=30$ , respectively. Through this transformation, all colors are representable by 7 root numbers of 2, 3, 5, 6, 10, 15, and 30. These root numbers encode millions of colors on a color wheel [10] .

Inspired by C₂₃₅, we intend to:

· find a set of integers which serve as the root of primes and twin primes,

· establish the Formula of Primes and build the Periodic Table of Primes (PTP) that allocates primes, and

· form the Counting Functions of Primes and Twin Primes.

Also accomplished include predicting within an interval, the largest k-tuple primes [8] with the same difference.

1.1. Steps and Notation Adopted to Build the PTP

First, we define a concise strategy by selecting the roots of primes greater than 10 and composites without factors of 2, 3, 5, and 7, and then form the Cyclic Table of Composites (CTC) by identifying the locations of such composites, followed by advancing the Formula of Primes and building the PTP. The procedures are summarized in 4 steps:

1) Selecting the roots and cycles

Adopting the first four primes 2, 3, 5, and 7, we take $2\times 3\times 5\times 7=210$ as the length of a period. Within the interval [11, 211], we sort out 48 integers of primes or composites that do not contain factors of 2, 3, 5, and 7. These 48 integers, to generate primes and composites, are considered as the roots denoted by $r_1=11$ , $r_2=13$ , $r_3=17$ , $\cdots$ , $r_{23}=103$ , $r_{24}=107$ , $\cdots$ , $r_{48}=211$ . These r_is are placed on the left column of a table.

2) Developing the Cyclic Table of Composites (CTC)

The CTC consists of multiple 48 × 48 tables $\left[l\left(i,j\right)\right]$ derived from the positions of composites without factors of 2, 3, 5, and 7.

Within an interval of $\left[48\left(\theta -1\right)+1,48\theta \right]$ , where $\theta \in N_{+}$ is considered a cycle, there are 48 integers which do not contain factors of 2, 3, 5, and 7. These $48\times \theta$ integers are either primes or composites, denoted as $q_1=r_1=11$ , $q_2=r_2=13$ , $q_3=r_3=17$ , $\cdots$ , $q_{48}=r_{48}=211$ , for cycle 1; $q_{49}=q_1+210=221$ , $q_{50}=q_2+210=223$ , $\cdots$ , $q_{96}=q_{48}+210=421$ , for cycle 2; $\cdots$ ; $q_{48\left(\theta -1\right)+1}=q_{48\left(\theta -2\right)+1}+210=q_1+210\left(\theta -1\right)$ , $q_{48\left(\theta -1\right)+2}=q_{48\left(\theta -2\right)+2}+210=q_2+210\left(\theta -1\right)$ , $\cdots$ , and $q_{48\theta }=q_{48\left(\theta -1\right)}+210=q_{48}+210\left(\theta -1\right)$ , for cycle θ. We call $q_1$ to $q_{48}$ segments of cycle 1, $q_{49}$ to $q_{96}$ segments of cycle 2, $\cdots$ , and $q_{48\times \left(\theta -1\right)+1}$ to $q_{48\theta }$ segments of cycle θ. The q_js are placed on the top row of a CTC table of θ cycles.

Formulating the CTC of the first cycle, namely, CTC(1), we observe the dual effect existing in $\left[l\left(i,j\right)\right]$ , where index i refers to $r_i$ and j refers to $q_j$ , such that in each row of this table, pairs of entries share the same value of elements $l\left(i,j\right)$ s. We also find for each row in the CTC(1), there is another row complementary to it, calling it the mirror effect. Specifically, there exist the mirror effects between the 23 top rows and the next 23 rows as well as between the 47^th row and the 48^th row of CTC(1). We define the basic CTC (CTC(basic)) to be a 24 × 48 table of $\left[l\left(i,j\right)\right]$ , $i=1,2,\cdots ,23,47$ and $j=1,2,\cdots ,\text{48}$ .

Utilizing the mirror effect, we find the complementary CTC (CTC (complementary)) a 24 × 48 table made of $\left[l\left(i,j\right)\right]$ for $i=24,25,\cdots ,\text{46},48$ and $j=1,2,\cdots ,\text{48}$ . Combining the CTC (basic) and the CTC(complementary), we form CTC(1).

We observe further an intermedia effect in CTC(1) between different cycles of composite tables where each is 48 × 48 in size.

3) Advancing the Formula of Primes

By deleting all composites located by CTC in an interval, we develop the Formula of Primes.

Given a positive integer b with b – 211 being a multiplier of 210, an integer $\alpha \in \left[0,b\right]$ without factors of 2, 3, 5, and 7 is a prime if and only if there exists an $i\in \left\{1,2,\cdots ,48\right\}$ such that $\alpha =r_i+210k$ , for some integer k with $k+1\notin L_b\left(i\right)$ , where $L_b\left(i\right)$ is a set of $l\left(i,j\right)$ s which satisfy some conditions of b and θ described in the Results section.

4) Building the Periodic Table of Primes (PTP)

Denote PTP( $k^0,k^{\prime }$ ) as a periodic table of primes from period $k^0$ to $k^{\prime }$ , $k^0<k^{\prime }\in N_{+}$ . PTP( $k^0,k^{\prime }$ ) is a $48\times \left(k^{\prime }-k^0+1\right)$ table composed of all primes within an interval $\left[a,b\right]$ , for $a=11+210\times k^0$ and $b=211+210\times k^{\prime }$ . Denote PTP⁺( $k^0,k^{\prime }$ ) as a table composed of primes in PTP( $k^0,k^{\prime }$ ) plus composites without factors 2, 3, 5, and 7 within the interval $\left[a,b\right]$ .

The steps of building a PTP are depicted in Figure 1. Starting from the CTC (initial), we form a CTC (initial dual). By combining the CTC (initial) and the CTC (initial dual), we obtain the CTC (basic) which is outlined in the Supplement under Establishing the CTC (basic) by the CTC (initial). Once CTC (1) is deduced from CTC (basic), all subsequent CTC (2), CTC (3), …, CTC (θ) are

Figure 1. Schematic process of forming the PTP.

readily available. Utilizing CTCs, we generate PTP⁺( $k^0,k^{\prime }$ ), where we delete all composites to obtain PTP( $k^0,k^{\prime }$ ). The red and green points in Figure 1 represent the diagonal and intermediate effects, respectively, and also show the cyclical pattern on CTC and the periodical pattern on PTP⁺.

1.2. Note

Lei et al. [11] held that the evolution of Chinese historical dynasties can be mapped by the properties of elements’ electrons in the Periodic Table of Chemical Elements. Wang [12] claimed that irregular primes are the survivors of natural numbers after regular decimations by primes.

There are various sieve methods to locate primes. In coming up with a solution for predicting primes, many have searched for a particular pattern of primes distributions. For example, Holt [13] mentioned structure to cycles due to recursion and created exact relative population models for all gaps up to a certain level. Dastych found a “mirroring” effect of numbers such as 2 × 3, 2 × 3 × 5 and 2 × 3 × 5 × 7 being represented by a wheel which were already in existence [14] . His algorithm originated from playing with Goldbach’s conjecture, but became useless when the numbers grew large. Others may have expressed their wishes to organize primes listing over the years. Although we tried to be thorough and are blessed by many professional colleagues in reaching this unique PTP, we could miss non-disclosed endeavors made by individuals beyond the publicly accessible domain.

2. Transformation and Observations

Without loss of generality, discussions in this section on the CTC, the CTC(basic) and the CTC(complementary) are referred to cycle 1, namely, CTC(1), unless stated otherwise.

In CTC, consisting of 48 rows and 48 columns, denote $r_i$ as the i^th row and $q_j$ the j^th column.

Let

$\begin{array}{c}{S}_1=\left\{r_1,r_2,\cdots ,r_{23},r_{47}\right\}\\ =\{11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,\\ 83,89,97,101,103,209\}\end{array}$

$\begin{array}{c}{S}_2=\left\{r_{24},r_{25},\cdots ,r_{46},r_{48}\right\}\\ =\{107,109,113,121,127,131,137,139,143,149,151,157,163,167,\\ 169,173,179,181,187,191,193,197,199,211\}\end{array}$

and

$S=S_1\cup S_2=\left\{r_1,r_2,\cdots ,r_{48}\right\}=\left\{q_1,q_2,\cdots ,q_{48}\right\}$ .

2.1. Transformation

Note that r_is and q_js are integers in $\left[11,211\right]$ without factors of 2, 3, 5, and 7. Let $\left(i,j\right)$ be at a position in the CTC and $l\left(i,j\right)$ the corresponding entry of point $\left(i,j\right)$ , where $1\le i,j\le 48$ , determined by

$l\left(i,j\right)=1+\left(q_j\times q_{\hat{j}}-r_i\right)/210$ . (1)

Given the i^th row and j^th column, we observe a unique $q_{\hat{j}}\in S$ that couples with $q_j$ such that $q_j\times q_{\hat{j}}-r_i$ is a multiplier of 210. Both $q_j$ and $q_{\hat{j}}$ depend on $r_i$ or the i^th row.

2.2. Three Statements

An outline of CTC is shown in Figure 2. Various effects in CTC are described by the following statements. Presented in Figure S1(a) along with an illustrative example are given in the Supplement under Example for Figure S1(a). Observing Figure S1(a), note that $l\left(i,i\right)=q_i+1$ , $i=1,2,\cdots ,23,47$ . We identify the diagonal effect in CTC in Statement 1 below.

Analyzing the distribution of $l\left(i,j\right)$ in the i^th row of Figure S1(a), we find the dual effect between $l\left(i,j\right)$ and $l\left(i,\hat{j}\right)$ , described in Statement 2 below.

Statement 2 implies that given $l\left(i,j\right)$ , we know $l\left(i,\hat{j}\right)$ , under which situation j and $\hat{j}$ are dual to each other. Take $i=1$ and $r_1=11$ for instance, $l\left(1,5\right)=l\left(1,8\right)=5$ , where $q_5\times q_8=23\times 37=851=11+210\left(5-1\right)$ . Notice that when $\hat{j}=j$ , $q_j^2$ is a descendant of $r_i$ and $l\left(i,j\right)$ appears in the i^th row only once. For instance, $l\left(48,9\right)=8$ appears only once and ${41}^2=1681=211+210\times \left(8-1\right)$ .

Further elaboration of the dual effect and associated examples are given in the Supplement under Elaboration of the Dual Effect of the CTC (basic). Analyzing Figure S1(a) and Figure S1(b), we also identify the mirror effect between each row of the CTC(basic) and its complementary row in CTC deduced below.

Figure 2. An outline of CTC(1).

Statement 3 implies that if we know $l\left(i,j\right)$ for $1\le i\le 23$ , then we can find $l\left(47-i,j\right)$ . Similarly, if we know $l\left(47,j\right)$ , then we find $l\left(48,j\right)$ . Following Statement 3, we form the CTC (complementary), the complementary of the CTC (basic).

CTC (complementary) is presented as Figure S1(b), in the Supplement. Merging the CTC (basic) and the CTC (complementary), we obtain the CTC (1) as the CTC for cycle 1, referring to Figure 2.

Consider CTC in Figure 2. There are two red diagonals shown in Figure 2, which further elaborates Figure S1(a) and Figure S1(b), The first diagonal is $l\left(1,1\right)=12$ , $l\left(2,2\right)=14$ , $\cdots$ , and $l\left(46,46\right)=200$ , and the second one is $l\left(46,1\right)=11$ , $l\left(45,2\right)=13$ , $\cdots$ , and $l\left(1,46\right)=199$ . The two cross lines in Figure 2 are obvious due to the mirror effect. $l\left(47,j\right)$ and $l\left(48,j\right)$ , $j=1,2,3,\cdots ,48$ , are mirrors with each other. An example is given in the Supplement under An Associated Example for Statement 3.

Figure S1(a) and Figure S1(b) show another distribution of $l\left(i,j\right)$ on CTC, named the intermedia effect, which happens at some symmetrical columns, described in the Supplement under Statement and Proof on the Intermedia effect in the CTC. Both the diagonal effect and the intermedia effect on the CTC are helpful in showing how to transfer CTC to PTP. In Figure 2, the red and green numbers represent the diagonal and the intermediate effects, respectively.

Proofs for the above Statements are given in the Supplement under Proofs of the Statements.

3. Results

This study provides three major results: the Formula of Primes, the Periodic Table of Primes, and the Counting Functions of Primes and Twin Primes.

Given a positive integer $b=211+210\times {\theta }^{*}{\hat{\theta }}^{*}$ , ${\theta }^{*},{\hat{\theta }}^{*}\in N_{+}$ , for $i\in \left\{1,2,\cdots ,48\right\}$ , define

$\begin{array}{l}{L}_b\left(i\right)\equiv \{l\left(i,j\right)+q_j\left(\hat{\theta }-1\right)+q_{\hat{j}}\left(\theta -1\right)+210\left(\theta -1\right)\left(\hat{\theta }-1\right)\le {\theta }^{*}{\hat{\theta }}^{*}+1\\ |j=1,2,\cdots ,48,\text{with}\text{thecorresponding}\hat{j}\text{defined}\\ \text{inExpression}\left(1\right),\theta =1,2,\cdots ,{\theta }^{*}\text{and}\hat{\theta }=1,2,\cdots ,{\hat{\theta }}^{*}\}\end{array}$ (2)

For the case ${\hat{\theta }}^{*}=1$ , then

$L_b\left(i\right)\equiv \left\{l\left(i,j+48\left(\theta -1\right)\right)\le {\theta }^{*}+1|j=1,2,\cdots ,48,\text{and}\theta =1,2,\cdots ,{\theta }^{*}\right\}$ (2’)

Notice that $L_b\left(i\right)$ is a set consisting of $l\left(i,j\right)\le {\Delta }_{i,\theta +1}$ , for $j=1,2,\cdots ,48\theta$ , where ${\Delta }_{i,\theta }={\min }_j\left\{l\left(i,j\right)|j=48\left(\theta -1\right)+1,48\left(\theta -1\right)+2,\cdots ,48\theta \right\}$ for $\theta \in N_{+}$ , and $210{\Delta }_{i,\theta +1}\ge b$ .

Also $l\left(i,j+48\left(\theta -1\right)\right)=l\left(i,j\right)+\left(\theta -1\right)q_{\hat{j}}$ , where $q_{\hat{j}}$ is derived from Expression (1). For any positive integer α containing no factors of 2, 3, 5, and 7, α must have a unique root $r_i\in S$ such that $\alpha -r_i$ is a multiple of 210. Moreover, if α is not a composite number, then it is a prime.

An example for $L_b\left(1\right)$ where $\hat{\theta }=1$ can be found in Supplement for the Text “7. Justification for Establishing $L_b\left(1\right)$ and Figure S2”.

3.1. The Formula of Primes

We summarize the Formula of Primes below:

A list of $L_b\left(1\right)$ for various b up to 44521 and $\hat{\theta }=1$ is given in Figure S2, which is presented in the Supplement.

3.2. The Periodic Table of Primes

According to CTC(θ) and the Formula of Primes, we specify the Periodic Table of Primes below.

Denote PTP⁺( $k^0,k^{\prime }$ ) as the table composed of primes in PTP( $k^0,k^{\prime }$ ) plus composites without factors of 2, 3, 5 and 7, within an interval $\left[11+211\times k^0,211\left(k^{\prime }+1\right)\right]$ .

Figure 3 is an outline of PTP⁺(0, 210), which includes PTP(0, 210) and composites without factors of 2, 3, 5, and 7 within the interval $\left[11,1+{210}^2\right]$ converted from Figure 2. Figure 3 has 48 rows and 211 columns, for $i=1,2,\cdots ,48$ , and $k=0,1,2,3,\cdots ,210$ . $\lambda \left(i,k\right)$ is an integer of the entry $\left(i,k\right)$ , computed from $l\left(i,j\right)$ of Figure 2. The two dotted red curves in Figure 3 are configured from two diagonal red lines of Figure 2, i.e., 12–14–18∙∙∙198–200 and 11–13–17∙∙∙ 197–199. The two dotted green diamond-shape lines in Figure 3 come from two green lines of Figure 2 of 24–75–74∙∙∙29–80 and 84–31–32∙∙∙77–24.

Figure 4 is a realization of PTP⁺(0, 10) including primes and composites without factors of 2, 3, 5, and 7. Further elaboration is seen in the Supplement under Numerical Illustrations for The Periodic Table of Primes. According to Gauss [8] [9] , the number of primes no more than b is approximately $b/\ln b$ . We find further the exact Counting Function of primes $\pi \left(b\right)$ when b – 211 is a multiplier of 210.

3.3. The Counting Functions of Primes and Twin Primes

We use $\pi \left(b\right)$ and ${\pi }^{*}\left(b\right)$ to represent the Counting Function of Primes and the Counting Function of Twin Primes in $\left[1,b\right]$ for a natural number b, respectively.

Currently, few predict the number of twin-prime pairs in an interval [2, b]. For b – 211 as a multiplier of 210, denote $T\left(i,i+1\right)$ as the set of twin-prime pairs on PTP( $0,\lceil b/211\rceil$ ), i.e., a set of k with $k+1\notin L_b\left(i\right)$ and $k+1\notin L_b\left(i+1\right)$ , where i and i+1 are for the i^th and the (i+1)^st row on the PTP, respectively. Our study finds the exact ${\pi }^{*}\left(b\right)$ below:

Figure 3. An outline of PTP⁺(0,210).

Figure 4. A partial table of PTP⁺(0,48).

Elaborations of Expressions (3) and (4) are given in the Supplement under Elaborations of Expressions (3) and (4), and examples of computing $\pi \left(b\right)$ and ${\pi }^{*}\left(b\right)$ are illustrated in the Supplement under Examples for the Predictions of Primes by the PTP.

4. Discussions

The unpredictability of prime numbers forms the basis of many applications, one being encryption called the RSA algorithm [1] . However, neither the ancient Sieve of Eratosthenes nor the modern Sieve of Atkin’s algorithms [15] have ever elaborated the physical meaning of finding primes. Torquato, Zhang and Courcy-Ireland [16] claimed that they found a physical structure pattern hidden in the distribution of prime numbers, but that discovery still didn’t explain the essence of prime numbers.

This paper identifies 48 natural numbers between 11 and 211, which do not contain factors of 2, 3, 5, and 7, to be the roots for generating all primes and composites without factors of 2, 3, 5, and 7. The locations of such composites exhibit periodic and cyclic properties, as represented by the CTC, which enable us to eliminate them for finding primes, as represented by the PTP. Treating the 48 roots as the genes of prime numbers, we can easily find the next prime of any given prime number and identify the next pair of twin primes. Our findings provide a platform to study many primes-related problems. No primes, twin primes or primes-related issues can ever surface if such issues are not rooted to the 48 integers. After all, prime numbers are not as random as many believe.

We form the CTC, followed by the PTP. All these present the primes effectively and with physical meaning. In addition, we can count the exact numbers of primes and twin primes within an interval. Discussed below are some further thoughts.

1) Instead of choosing 2, 3, 5 and 7, one may add 11, 13, or more primes to generate the roots. By so doing, the PTP will gradually become gigantic, too complicated, and too difficult to visualize, although likely more effective. If one is interested in the behaviors of super-large primes, one could find it useful in such large tables, which may be extended to infinite. On the other hand, one may choose 2, 3, and 5 to generate 8 roots for a small prime table.

2) The PTP is helpful in understanding many unclear phenomena. For instance, it explains a troublesome observation [5] that for a given prime with the last digit of 1, the chance of its next prime to have 1 as its last digit is much less than that of 3 or 7 or 9. From the Formula of Primes, if a given prime is 221 (i.e., $\theta =2$ , $r_i=11$ ), then the most possible near primes should be firstly 223 (i.e., $\theta =2$ , $r_i=13$ ), followed by 227, 229, 233, 239, and 241.

3) From the 48 roots identified, we find no triplet or higher multiples of primes existing in the roots. Therefore, there will be no triplet primes found in future generations. Likewise, all twin primes will appear exactly at the parallel locations as those appearing in the 48 roots of the PTP. This implies that all twin-prime pairs are descendants of 15 pairs of twin primes or composites. In fact, an equal chance is found in the last digit of the 48 roots for 1, 3, 7, and 9. Therefore, the last digits for all primes will each have a 25% chance of being 1, 3, 7, and 9 when primes go to infinity.

4) This is the first time that a visualizable prime table is built with proofs using a manageable set of primes as the basis for making useable influences and clarifying some questions of interest in existence for years. Just like reported, openly or not, by Oliver and Soundaranjan [4] , Wang [12] , Holt [13] , Dastych [14] , etc., no conclusion has been drawn on establishing a compact table for primes of any kind.

In contrast, from building the CTC, we demonstrate that every prime has an ancestor among 48 integers which include 43 primes and 5 composites, and every twin-prime pair comes from an ancestor of 15 pairs of these 48 integers. We develop the algorithms and give illustrating examples in the Supplement using the PTP from which we draw several inferences and present some useful applications.

5) Beyond the fundamental investigation, this study is due to part of our efforts in exploring various applications of primes, including systems reliability design [2] , and building a color system C₂₃₅ to unify RGB and CMYK and to encode millions of colors on a color wheel [10] . The universal color system wins the Special Prize and the Gold Medal with Congratulations of the Jury at the 49^th International Exhibition of Inventions of Geneva.

5. Conclusions

This paper selected 48 integers as the roots to generate primes and composites without the factors of 2, 3, 5, and 7. We constructed a composite table, CTC, further observing the diagonal effect, the dual effect and the mirror effect. Based on the CTC, this paper introduced the first closed form expressions for the Formula of Primes and the Periodic Table of Primes. The Counting Functions of Primes and Twin Primes are then readily deduced. Related mathematical proofs and computations are exercised to testify the correctness of the above statements.

While there exist proven and unproven concepts, approaches and analysis in the literature, our study clearly shows that any prime except 2, 3, 5, and 7 can be uniquely rooted to one of 48 natural numbers between 11 and 220 in cycles of length 210. Moreover, any twin-prime pair can be uniquely rooted to a pair of the same 48 roots in cycles of length 210. We show that all composite numbers with no factors of 2, 3, 5, and 7 appear in cyclic manner. To the best of our knowledge, these fundamental findings are the first to systematically and concretely address in the open literature. People may have used terminologies similar to this study, but neither similar results nor close form solutions have ever been presented.

Acknowledgements

We thank Nianrui Lin of CityU for thorough review of the article and verifying all the examples with valuable inputs. We also acknowledge more than two dozens of colleagues and reviewers in mathematics and computer science for comments and providing the suggestions. Early version of the article has been archived at SSRN.

Supplement for the Text

1. Establishing the CTC (basic) by the CTC (initial)

Specifically, we adopt the following steps to establish the CTC (basic):

a) From the CTC (basic) of Figure 1, we find 23 × 23 CTC (initial) table for the first cycle. In the CTC (initial), its entries of $\left[l\left(i,j\right)\right]$ , for $i=1,2,\cdots ,23$ and $j=1,2,\cdots ,23$ , are both starting from 11; namely, the left column contains $r_1,r_2,\cdots ,r_{23}$ and the top row contains $q_1,q_2,\cdots ,q_{23}$ .

b) Applying the dual effect, we form another 23 × 23 CTC (initial dual) table from CTC (initial). In this new table, its entries of $\left[l\left(i,j\right)\right]$ , for $i=1,2,\cdots ,23$ and $j=24,25,\cdots ,46$ , are the first 23 roots starting from 11 for “i”s and the 24^th to 46^th roots for “j”s, respectively. Namely, the left column contains $r_1,r_2,\cdots ,r_{23}$ , and the top row contains $q_{24},q_{25},\cdots ,q_{46}$ .

c) Add r₄₇ as the end of the 24^th row and add q₄₇ and q₄₈ as the end of the 47^th and the 48^th columns. Namely, the left column contains r₄₇ as the last index and the top row contains q₄₇, q₄₈ as the last two indices.

d) Combine tables developed by a, b, and c steps to complete CTC(basic).

As shown in Figure 1, there are the CTC(initial), the CTC(initial-dual), the CTC(basic), the CTC(complementary), and CTC(1).

$\text{CTC}\left(\text{initial}\right)=\{l\left(i,j\right)|i=1,2,\cdots ,23;j=1,2,\cdots ,23\}$ ,

$\text{CTC}\left(\text{initial-dual}\right)=\{l\left(i,j\right)|i=1,2,\cdots ,23;j=24,\cdots ,46\}$ ,

$\text{CTC}\left(\text{basic}\right)=\{l\left(i,j\right)|i=1,2,\cdots ,23,47;j=1,2,\cdots ,48\}$ ,

$\text{CTC}\left(\text{complementary}\right)=\{l\left(i,j\right)|i=24,25,\cdots ,46,48;j=1,2,\cdots ,48\}$ ,

$\text{CTC}\left(1\right)=\{l\left(i,j\right)|i,j=1,2,\cdots ,48\}$ .

2. Example for Figure S1(a)

Given $q_1=11$ , compute $l\left(1,1\right)$ as follows:

Since $l\left(1,1\right)=1+\frac{q_1\times q_{\hat{1}}-r_1}{210}$ , and the unique $q_{\hat{1}}$ to let $\frac{11\times q_{\hat{1}}-11}{210}$ be an integer is

$q_{\hat{1}}=211,l\left(1,1\right)=1+\frac{11\times 211-11}{210}=1+11=12.$

Similarly, we find

$l\left(2,1\right),l\left(3,1\right),l\left(4,1\right),l\left(5,1\right),l\left(6,1\right),\cdots ,l\left(23,1\right),l\left(47,1\right)=\left(10,6,4,11,5,\cdots ,8,1\right)$ .

3. Elaboration of the Dual Effect of the CTC (basic)

By utilizing Statement 2, we can generate a new table CTC(initial dual) in the Introduction from CTC(initial). Taking the first row, i.e. $i=1$ and $r_1=11$ as an example (Figure 2):

(i) Since $l\left(1,2\right)=l\left(1,3\right)=2$ , we have

$l\left(1,47-2\right)=l\left(1,47-3\right)=1+\frac{q_{45}\times q_{44}}{q_2\times q_3}\left(\frac{11}{210}+l\left(1,2\right)-1\right)-\frac{11}{210}=182$

Similarly, since $l\left(1,5\right)=l\left(1,8\right)=5$ , we have

$l\left(1,47-5\right)=l\left(1,47-8\right)=1+\frac{q_{42}\times q_{39}}{q_5\times q_8}\left(\frac{11}{210}+l\left(1,5\right)-1\right)-\frac{11}{210}=155$

(ii) For $i=1$ and $j=1,6,10,12,\cdots ,19,21,22$ , we do not have $1\le \hat{j}\le 23$ to fit $l\left(1,j\right)=l\left(1,\hat{j}\right)$ , then we compute $l\left(1,\hat{j}\right)$ for $24\le \hat{j}\le 46$ .

When $j=1$ , $q_1=11$ and $l\left(1,1\right)=12$ ,

since $\frac{r_1+210\left(l\left(1,1\right)-1\right)}{q_1}=\frac{11+210\left(12-1\right)}{11}=211=q_{48}$ , we have

$\hat{1}=48$ and $l\left(1,48\right)=l\left(1,1\right)=12$

When $j=6$ , $q_6=29$ and $l\left(1,6\right)=16$ ,

since $\frac{r_1+210\left(l\left(1,6\right)-1\right)}{q_6}=109=q_{25}$ , we have

$\hat{6}=25$ and $l\left(1,25\right)=l\left(1,6\right)=16$ .

Similarly, we can predict $l\left(1,26\right)=l\left(1,15\right)=37$ , $l\left(1,28\right)=l\left(1,12\right)=33$ , $l\left(1,29\right)=l\left(1,14\right)=39$ , $l\left(1,30\right)=l\left(1,10\right)=29$ , $\cdots$ , $l\left(1,41\right)=l\left(1,22\right)=88$ .

4. An Associated Example for Statement 3

Comparing the CTC (basic) with the CTC (complementary) and referring to Figure S1(a) and Figure S1(b), it is clear that:

Given $j=1$ and $q_1=11$ , we have $l\left(1,1\right)+l\left(46,1\right)=12+11=2q_1+1$ .

Given $j=2$ and $q_2=13$ , we have $l\left(2,2\right)+l\left(45,2\right)=14+13=2q_2+1$ .

Given $j=1$ and $q_1=11$ , for $i=2$ , we have $l\left(45,1\right)+l\left(2,1\right)=2+10=12=q_1+1$ .

Given $j=12$ and $q_{12}=53$ , for $i=1$ , $l\left(46,12\right)+l\left(1,12\right)=21+33=54=q_{12}+1$ ; for $i=23$ , $l\left(24,12\right)+l\left(23,12\right)=28+26=54=q_{12}+1$ ; for $i=47$ , $l\left(47,12\right)+l\left(48,12\right)=26+27=53=q_{12}$ .

5. Statement and Proof of the Intermedia Effect in the CTC(1)

Figure S1(a) and Figure S1(b) show another distribution of $l\left(i,j\right)$ on the CTC, named the intermedia effect, which happens at some symmetrical columns, described below.

For instances, in the case of green columns 23 and 24 of Figure 2, $l\left(19,23\right)=l\left(28,24\right)=6$ , $\cdots$ , $l\left(2,23\right)+2=l\left(45,24\right)=77$ . Other column pairs also exhibit the intermedia effect. For example, in the case of columns 22 and 25, if $l\left(i,22\right)+l\left(i,25\right)=110$ , then $l\left(i,22\right)=l\left(47-i,25\right)$ .

Proof of the intermedia effect in the CTC

For the case of columns 23 and 24,

(i) From the mirror effect, we have $l\left(i,24\right)+l\left(47-i,24\right)=q_{24}+1=108$ .

If $l\left(i,23\right)+l\left(i,24\right)=108$ , then $l\left(i,23\right)=108-l\left(i,24\right)=l\left(47-i,24\right)$ .

(ii) If $l\left(i,23\right)+l\left(i,24\right)=106$ , then $l\left(i,23\right)+2=108-l\left(i,24\right)=l\left(47-i,24\right)$ .

For the case of columns 22 and 25, from the mirror effect, we know $l\left(i,25\right)+l\left(47-i,25\right)=q_{25}+1=109+1=110$

If $l\left(i,22\right)+l\left(i,25\right)=110$ , then $l\left(i,22\right)=l\left(47-i,25\right)$ .

6. Proofs of the Statements

Proof of Statement 1

For each $i,j=1,2,\cdots ,48$ , notice that in CTC(1),

$\begin{array}{c}l\left(j,j\right)=1+\frac{q_j\times q_{\hat{j}}-r_j}{210}\\ =1+\frac{q_j\left(211-1\right)}{210}\\ =1+q_j,\\ \text{as}{r}_j=q_j\text{and}\text{we}\text{choose}{q}_{\hat{j}}=q_{48}=211.\end{array}$

For a given j, $r_i=q_j\times q_{\hat{j}}-\left(l\left(i,j\right)-1\right)\times 210$ ,for some $\hat{j}$ .This means that a fixed $r_i$ is the remainder of $q_j\times q_{\hat{j}}$ subtracting the maximum multiple of 210 smaller than $q_j\times q_{\hat{j}}$ . Since $q_{\hat{j}}=q_{48}=211$ is the largest $q_{\hat{j}}$ hence $l\left(j,j\right)\ge l\left(i,j\right)$ .

Proof of Statement 2

By definition, $l\left(i,j\right)=1+\frac{q_j{q}_{\hat{j}}-r_i}{210}$ , $l\left(i,\hat{j}\right)=1+\frac{q_{\hat{j}}{q}_j-r_i}{210}$ .

Therefore, we have $l\left(i,j\right)=l\left(i,\hat{j}\right)=1+\frac{q_j\times q_{\hat{j}}-r_i}{210}$

Proof of Statement 3

(i):

In this proof, we use $q_{\hat{j}|i}$ to differentiate $q_{\hat{j}}$ given different $r_i$ .

Notice that $r_{47-i}+r_i=210$ , $\forall i=1,\cdots ,23$ ,

$l\left(47-i,j\right)=1+\frac{q_{j|i}{q}_{\hat{j}|\left(47-i\right)}-r_{47-i}}{210}$ for some $q_{\hat{j}|47-i}$ and

$l\left(i,j\right)=1+\frac{q_{j|i}{q}_{\hat{j}|i}-r_i}{210}$ forsome $q_{\hat{j}|i}$ .

Hence,

$\begin{array}{c}l\left(47-i,j\right)+l\left(i,j\right)=2+\frac{q_{j|i}\left(q_{\hat{j}|\left(47-i\right)}+q_{\hat{j}|i}\right)-210}{210}\\ =1+\frac{q_{j|i}\left(q_{\hat{j}|\left(47-i\right)}+q_{\hat{j}|i}\right)}{210}\end{array}$

Since $\frac{q_{j|i}\left(q_{\hat{j}|\left(47-i\right)}+q_{\hat{j}|i}\right)}{210}$ must be an integer, and $q_{j|i}$ has no factors of 2, 3, 5, 7, $q_{\hat{j}|\left(47-i\right)}+q_{\hat{j}|i}$ must be a multiple of 210. This happens only when

$q_{\hat{j}|\left(47-i\right)}+q_{\hat{j}|i}=\{\begin{array}{l}210,\text{if}\hat{j}|\left(47-i\right)+\hat{j}|i=47\\ 420,\text{if}\hat{j}\left|\left(47-i\right)\text{or}\hat{j}\right|i=48\end{array}$

Therefore,

$l\left(47-i,j\right)+l\left(i,j\right)=\{\begin{array}{l}{q}_{j|i}+1,\text{if}i\ne j\text{and}47-i\ne j\\ 2q_{j|i}+1,\text{if}i=j\text{or}47-i=j\end{array}$

(ii):

Notice that $r_{47}+r_{48}=209+211=420$ . Hence

$\begin{array}{c}l\left(47,j\right)+l\left(48,j\right)=\left(1+\frac{q_{j|47}{q}_{\hat{j}|47}-r_{47}}{210}\right)+\left(1+\frac{q_{j48}{q}_{\hat{j}|48}-r_{48}}{210}\right)\\ =\frac{q_{j|i}\left(q_{\hat{j}|47}+q_{\hat{j}|48}\right)}{210}\\ =\{\begin{array}{l}{q}_{j|i},\text{if}\hat{j}|47+\hat{j}|48=47\\ 2q_{j|i},\text{if}\hat{j}\left|47\text{or}\hat{j}\right|48=48\end{array}\end{array}$

The 2^nd case happens only when $j=47,48$ .

7. Justification for Establishing L_b(1) and Figure S2

Starting with $\theta =1$ , compute $q_j,q_{\hat{j}}$ and $l\left(1,j\right)$ using Expression (1)

$l\left(1,j\right)=1+\frac{q_j\times q_{\hat{j}}-11}{210}$ , for $j=1,2,\cdots ,48$ .

Then for $2\le \theta \le 19$ and $\hat{\theta }=1$ , we have:

$l\left(1,j+48\right)=l\left(1,j\right)+q_{\hat{j}}$

$l\left(1,j+96\right)=l\left(1,j\right)+2q_{\hat{j}}$

$⋮$

$l\left(1,j+864\right)=l\left(1,j\right)+18q_{\hat{j}}$

We then list $l\left(i,j+48\left(\theta -1\right)\right)$ for $\theta =1,2,3,\cdots ,19$ in Figure S2, corresponding to Expression (2’). The bottom row of Figure S3 demonstrates that ${\Delta }_{1,1}=2$ , ${\Delta }_{1,2}=15$ , ${\Delta }_{1,3}=28$ , $\cdots$ , ${\Delta }_{1,19}=210$ . Notably,

${\min }_j\left\{l\left(i,j+48\left(\theta -1\right)\right)\right\}<{\min }_j\left\{l\left(i,j+48\theta \right)\right\}$ for all $\theta =1,2,3,\cdots ,19$ , and ${\Delta }_{i,1}<{\Delta }_{i,2}<{\Delta }_{i,3}<\cdots$ , for all i.

For instance, given $b=2311=211+210\times 10$ , compute $L_{2311}\left(1\right)$ by applying Expression (2)to obtain $L_{2311}\left(1\right)=\left\{2,5,7,9\right\}$ .

According to the Formula of Primes, in order for $\alpha =11+210k\le 2311$ to be primes, it requires $k+1\notin L_{2311}\left(1\right)$ for $k\in \left\{1,2,\cdots ,10\right\}$ . Therefore, we let

$k\in \left\{0,2,3,5,7,9,10\right\}$ and get the following primes: 11, $11+210\times 2=431$ , $11+210\times 3=641$ , $11+210\times 5=1061$ , $11+210\times 7=1481$ , $11+210\times 9=1901$ and $11+210\times 10=2111$ . They are all shown in the first row of Figure 4.

Given $b=211+210\times 28=6091$ , we get

$L_{6091}\left(1\right)=\left\{2,5,7,9,12,15,16,17,19,23,24,28\right\}$ .

In order for $\alpha =11+210k\le 6091$ to be primes, we let $k\in \left\{0,2,3,5,7,9,10,12,13,17,19,20,21,24,25,26\right\}$ such that $k+1\in L_{6091}\left(1\right)$ to obtain the following primes: 11, 431, 641, 1061, 1481, 1901, 2111, 2531, 2741, 3581, 4001, 4211, 4421, 5051, 5261, 5471.

8. Elaborations of Expressions (3) and (4)

Proof of Expression (3)

The total number of elements in PTP⁺(0, k) is $48\times \left(k+1\right)$ , for $k\in N_0$ . Within the interval $\left[11,{211}^2\theta \right]$ , the number of different $l\left(i,j\right)$ is $\left|L_b\left(i\right)\right|$ for $b=211+210k$ . Hence, the total number of composites without 2, 3, 5, and 7 within the above interval is $\left|L_b\left(i\right)\right|$ . Note that 4 primes are less than 10. Therefore, the number of primes within $\left[2,b\right]$ is

$\pi \left(b\right)=4+48\times \left(k+1\right)-{\displaystyle {\sum }_{i=1}^{48}\left|L_b\left(i\right)\right|}$ .

Proof of Expression (4)

For an integer $i\in W=\{1,3,6,9,13,\cdots ,45,47\}$ , $(r_i,r_{i+1})$ are twin primes for $r_i\ge 11$ . If there exists a k, $k\in N_0$ , such that $k+1\notin L_b\left(i\right)$ and $k+1\notin L_b\left(i+1\right)$ , then $r_i+210k$ and $r_{i+1}+210k$ are twin primes. Since (2, 3), (3, 5), (5, 7) are the only three twin-prime pairs less than 11. Therefore, the number of twin-prime pair in interval $\left[2,b\right]$ is ${\pi }^{*}\left(b\right)=3+{\displaystyle {\sum }_{i\in W}\left|T\left(i,i+1\right)\right|}$ , where $T\left(i,i+1\right)$ is a set of k such that $k+1\notin L_b\left(i\right)$ and $k+1\notin L_b\left(i+1\right)$ .

9. Numerical Illustrations for The Periodic Table of Primes

Let $\lambda \left(i,k\right)$ be the integer value of the entry $\left(i,k\right)$ in PTP⁺(0, 210) if Figure 3, which is computed based on CTC (1) and the Formula of Primes. For instance, $\lambda \left(i,k\right)$ on the diagonal dotted points (1, 11), (2, 13), (3, 17), (4, 19) and (5, 23) are computed below:

Now return to columns 22 and 25 in Figure 2. We note that (i, 23) and (i, 24), for $i=1,2,\cdots ,48$ being the intermediate dotted points. There are two other green lines 12–10–9–7–6 for column 22, and 6–7–9–10–12 for column 25. Both are caused by the intermedia effect, with respect to 47. These two lines are also converted to Figure 4, shown as 11 × 101 – 13 × 101 – 17 × 101 – 19 × 101 – 23 × 101, and 11 × 109 – 13 × 109 – 17 × 109 – 19 × 109 – 23 × 109, respectively.

The above conversions between $l\left(i,j\right)$ in the CTC and $\lambda \left(i,k\right)$ in the PTP⁺ demonstrate that (i) many composites such as the digonal and intermediate dotted points which allocate cyclically at the CTC are also allocated periodically at the PTP⁺. (ii) the PTP is obtained by removing all composites periodically allocated at PTP⁺. Therefore, we claim that prime numbers are also generated periodically since they come from removing all periodically distributed composites.

Figure 4 contains the first 11 columns for the PTP⁺(0, 210), which is composed of 48 × 11 = 528 elements, represented by $\lambda \left(i,k\right)$ and elaborated below.

(i) Each entry $\lambda \left(i,k\right)$ is either a prime or a composite.

(ii) If $\lambda \left(i,k\right)$ is a prime, then $\lambda \left(i,k\right)=r_i+210k$ , and vice versa. If $\lambda \left(i,k\right)$ is a composite, then there are $q_j$ and $q_{\hat{j}}$ such that

$\lambda \left(i,k\right)=q_j\times q_{\hat{j}}=r_i+210\times \left(l\left(i,j\right)-1\right)$ .

Take $i=1$ , $r_i=11$ for instance,

$\lambda \left(1,1\right)=11+210\times \left(l\left(1,2\right)-1\right)=q_2\times q_3=13\times 17$ ,

$\lambda \left(1,4\right)=11+210\times \left(l\left(1,5\right)-1\right)=q_5\times q_8=23\times 37$ .

Then, we have $\lambda \left(1,0\right)=11$ , $\lambda \left(1,2\right)=11\times 210\left(3-1\right)=431$ ,

$\lambda \left(1,3\right)=11+210\times \left(4-1\right)=641$ , $\cdots$ ,

$\lambda \left(1,9\right)=11+210\times \left(10-1\right)=1901$ .

(iii) Primes $\lambda \left(i,k\right)$ and ${\lambda }^{\prime }\left(i,k\right)=\lambda \left(i,k\right)+1$ are twin-prime pairs, if and only if ${r^{\prime }}_i=2+r_i$ , where $\lambda \left(i,k\right)=r_i+210\times k$ , ${\lambda }^{\prime }\left(i,k\right)={r^{\prime }}_i+210\times k$ .

For instance, (431, 433), (641, 643), (1061, 1063), (1481, 1483) are twin-prime pairs.

(iv) The numbers of primes in column k of the PTP⁺(0, 210) are shown as (43, 35, 32, 31, 31, 28, 28, 30, 26, 27, ⋯). The sum of primes within interval [11, 2311] is 43 + 35 + 32 + ⋯+ 18 = 329. The number of twin-prime pairs of a column k are listed as (12, 6, 4, 4, 4, 8, 6, 4, 4, 4, ⋯, 3).The maximum prime gap for the k^th column can also be found from Figure 4. For instance, the maximum prime gap for column 2 is 10, occurring at 409 and 419 of r₄₆ and r₄₇, respectively.

(v) The PTP(k⁰,k') is useful in predicting primes. Examples are given by Examples for the Predictions of Primes by the PTP.

In summary, the Formula of Primes, for the case $\hat{\theta }=1$ , is checked below:

From Figure S1(a), we know that

$L_b\left(1\right)=\{2,5,7,9,12,\cdots \}$

$L_b\left(2\right)=\{7,10,12,13,14,\cdots \}$

$L_b\left(3\right)=\{3,6,11,12,13,14,\cdots \}$

⋮

From Figure 4, we know that

$\alpha =r_1+210k$ is prime for $k=0,2,3,5,7,9,10$

$\alpha =r_2+210k$ is prime for $k=0,1,2,3,4,5,7,8,10$

$\alpha =r_3+210k$ is prime for $k=0,1,3,4,6,7,8,9$

⋮

In general, $\alpha =r_i+210k$ is prime, if $k+1\notin L_b\left(i\right)$ and $\hat{\theta }=1$ .

10. Examples for the Predictions of Primes by the PTP

(a)(b)

Figure S1. (a): A basic composite table CTC (basic); (b): A complementary composite table CTC (complementary).

Figure S2. List of L₄₄₅₂₁(1), corresponding to Expression (2’), for θ = 1, 2, …, 19.

Figure S3. The periodic table of primes for various intervals.

Conflicts of Interest

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

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