Han-Lin Li¹, Shu-Cherng Fang², Way Kuo³, Nianrui Lin^4
1Department of Computer Science, City University of Hong Kong, Hong Kong SAR, China.
²Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, USA.
³Hong Kong Institute for Advanced Study and Department of Data Science, City University of Hong Kong, Hong Kong SAR, China.
⁴Department of Systems Engineering, City University of Hong Kong, Hong Kong SAR, China.
DOI: 10.4236/apm.2026.161006   PDF    HTML   XML   130 Downloads   523 Views   Citations

Abstract

Traditionally, primes have been viewed as weeds appearing irregularly among the natural numbers, making their locations difficult to predict. We believe composite numbers, the complement of primes, exhibit symmetric patterns in their distribution. This study presents a new perspective: after systematically eliminating integers divisible by small primes such as 2, 3, 5, and 7, the remaining integers can be uniquely represented through a framework of roots and kins. Furthermore, all remaining composite numbers exhibit cyclic and mirror effects, enabling the construction of an extended Cyclic Table of Composites. Using this table, we derive a formula that identifies the locations of primes within any interval. Additional formulas for fast factorization and for locating twin primes, mirror primes, and prime tuples are also obtained. These results suggest that although primes themselves appear structureless, the composites surrounding them obey rich internal symmetries that allow efficient detection of “where primes hide”.

Keywords

Primes, Composites, PTP, CTC, eCTC

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1. Introduction

Leonhard Euler once remarked, “Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate” [1]. Don Zagier similarly noted that prime numbers “grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout” [2]. In fact, a few patterns have been observed in prime distribution, one of which is that prime numbers demonstrate higher density on the diagonal, horizontal, and vertical lines of the Ulam spiral (equivalently, certain quadratic polynomials generate more primes than others), but with no rigorous proof [3]. Even today, the fundamental question remains: given an interval of integers, where do the primes hide? In other words, we lack an explicit, efficiently computable rule for locating primes inside an arbitrary interval.

Classical sieve methods locate primes by removing all composites, but provide only partial solutions. The ancient Sieve of Eratosthenes [4] and its variants (e.g., Augmented Eratosthenes [4], Segmented Sieve of Eratosthenes [5], and Sieve of Sundaram [6]) exhaustively enumerate composites of certain multiplicative forms. Sieve of Atkin and Bernstein [7] leverages the properties of binary quadratic forms to recognize composites. Although the wheel sieve method [8] provides insight into the cyclic feature of composite distribution, it didn’t consider the symmetrical patterns of composites.

The number theory literature [1] [2] [4]-[11] affirms that no existing method provides a direct, effectively computable formula for listing primes solely within a given interval. Gauss’s prime number theorem [9] gives a celebrated asymptotic approximation of how many primes lie in an interval, but it cannot tell which values are prime.

Primes play essential roles in data science, color theory [12], reliability engineering [13], and cryptography. Our previous works introduced the Periodic Table of Primes (PTP) [14], periodic prime listings [15], and kernel-based semiprime factorization [16]. In this study, we uncover a striking phenomenon: once small primes are filtered out, the remaining composites form a structure governed by strong horizontal, vertical, and diagonal mirror symmetries. These symmetries allow rapid identification of composites within an interval, and the primes are then simply the remaining entries.

Our main contributions include: 1) Discovery and confirmation of the mirror symmetry in composite numbers, 2) Introduction of the kin-root framework, which shows that although primes appear structureless, composites possess strong governing structure, 3) Development of an extended Cyclic Table of Composites that enables efficient local processing of intervals, and 4) Formulas for prime location, factorization, twin primes, Pythagorean primes, prime tuples, and neighboring primes. This study offers an extension of the method to larger moduli for identifying large primes and factoring large composites.

2. Preliminaries

The Periodic Table of Primes was first derived from the Cyclic Table of Composites (CTC) by Li, Fang, and Kuo [14], which includes composite numbers without factors of 2, 3, 5, and 7 up to 211². Though a similar period ($2\times 3\times 5\times 7=210$ ) appeared in the wheel sieve method [8], CTC is derived through a novel, rigorous method. In this paper, we reorganize the composites and reconstruct the CTC from a different angle, revealing the symmetry properties of the composites.

Our study is based on the following statements, which can be readily verified from [14]-[16].

Statement 1

Let

${\lambda }_7=2\times 3\times 5\times 7=210$ .

We can construct a set $S_7$ containing all integers less than 210 that are not divisible by 2, 3, 5, or 7. This set contains 48 such integers:

$S_7=\left\{11,13,17,23,\cdots ,103,107,\cdots ,199,209,211\right\}$ .

Label these integers as

$q_1=11$ , $q_2=13$ , $q_3=17$ , $\cdots$ , $q_{23}=103$ , $q_{24}=107$ , $\cdots$ , $q_{46}=199$ , $q_{47}=209$ , $q_{48}=211$ .

Statement 2

For any positive integer $\alpha$ that contains no factors of 2, 3, 5, or 7, $\alpha$ can be expressed uniquely as

$\alpha =r_i+{\lambda }_{7}k=r_i+210k$ ,

where $r_i\in S_7$ is called the root, k is an integer (the kin), 210 is the period, and $0\le k\le 210$ . Thus, all such integers fall into 48 congruence classes modulo 210.

Statement 3

Every such integer $\alpha$ is either prime or composite. If $\alpha$ is composite, then it can be written as

$\alpha =r_i+210k=q_{j}m$ ,

where $q_j\in S_7$ divides $\alpha$ and m is a positive integer.

Statement 4

To identify all composites and primes within an interval, it suffices to analyze the canonical interval

$\left[11+210k,211+210k\right]$ , $0\le k\le 210$

Once all composites within this interval are listed, the primes in the same interval follow immediately as the remaining values.

Statement 5

Composites sharing the same root and factor exhibit a cyclic pattern. Let

$\alpha =r_i+210k=q_{j}m$ ,

and consider

${\alpha }^{\prime }=r_i+210\left(k+q_j\right)$ .

Then, ${\alpha }^{\prime }$ has a kin value that is $q_j$ cycles larger than that of $\alpha$ , and satisfies

${\alpha }^{\prime }=q_j\left(m+210\right)$ .

Thus, composites repeat periodically at intervals proportional to their factor $q_j$ .

Statement 6

Each composite

${\alpha }_0=r_i+210k=q_{j}m$

has three corresponding “mirror” composites:

Vertical mirror:

${\alpha }_1=-{\alpha }_0+210q_j$ .

Horizontal mirror:

${\alpha }_2=-{\alpha }_0+210m$ .

Diagonal mirror:

${\alpha }_3={\alpha }_0+210\times \left(210-q_j-m\right)$ .

These mirror relationships create a symmetrical structure among composites. See Theorem 2 for further details.

Statement 7

Using the cyclic and mirror effects, we construct a 210-based extended Cyclic Table of Composites (eCTC), denoted eCTC₂₁₀, containing tuples of the form

$\left(k_j,t_i\right)$ .

This table allows rapid identification of all composites in any interval.

Statement 8

With the eCTC, we derive:

• a formula for locating primes efficiently within an interval,

• a factorization formula based on kin-root relationships, and

• a tuple-prime formula for identifying prime pairs and higher-order prime clusters.

Statement 9

Although Statements 1-8 are based on ${\lambda }_7=210$ , the method extends naturally to larger moduli. For example:

${\lambda }_{11}=2\times 3\times 5\times 7\times 11=2310$ ,

${\lambda }_{13}=2\times 3\times 5\times 7\times 11\times 13=30030$ .

One may construct:

• a kin-root table eCTC₂₃₁0 to find primes in

$\left[13+2310k,2311+2310k\right]$ .

• or a kin-root table eCTC₃₀₀₃₀ to find primes in

$\left[17+30030k,30031+30030k\right]$ .

These generalizations allow the method to scale to increasingly large intervals.

3. Extended Cyclic Table of Composites

In this section, we construct the proposed eCTC and examine its underlying structural properties.

Let

${\lambda }_x={\displaystyle \prod }_{p\in \left\{2,3,5,\cdots ,x\right\}}p$

be the product of all primes from 2 through x. For example:

${\lambda }_7=2\times 3\times 5\times 7=210$ , ${\lambda }_{11}=2310$ , ${\lambda }_{13}=30030$ , ${\lambda }_{17}=510510$ .

3.1. Definition of the Root Set S_x

Let $S_x$ denote the set of positive integers not exceeding ${\lambda }_x+1$ that have no prime factors among $2,3,5,\cdots ,x$ . Examples include:

• $S_7=\left\{11,13,17,19,23,27,\cdots ,211\right\}$ .

• $S_{11}=\left\{13,17,19,23,27,\cdots ,2311\right\}$ .

• $S_{13}=\left\{17,19,23,27,29,\cdots ,30031\right\}$ .

Let |$S_x$ | denote the number of elements in $S_x$ . For instance, $\left|S_7\right|=48$ . In this section, we focus on the case $x=7$ , i.e., ${\lambda }_7=210$ .

3.2. Kin-Root Representation

Consider any integer $\alpha$ satisfying

$11\le \alpha \le {211}^2$ and $\text{gcd}\left(\alpha ,210\right)=1$ .

Then, $\alpha$ can be uniquely expressed as:

$\alpha =r_i+210k$ , $i\in \left\{1,2,3,\cdots ,\left|S_7\right|\right\}$ , $0\le k\le 210$ . (1)

where

• $r_i\in S_7$ is the root,

• 210 is the period,

• k is the kin value of $\alpha$ . In other words, $\alpha$ is the k-th generation of $r_i$ .

Thus, $\left(k,r_i\right)$ is the kin-root representation of $\alpha$ , serving as the key element in the table proposed later.

The full set $S_7$ is:

$\begin{array}{l}{S}_7=\{r_1=11,r_2=13,r_3=17,r_4=19,r_5=23,r_6=29,r_7=31,\\ r_8=37,r_9=41,r_{10}=43,r_{11}=47,r_{12}=53,r_{13}=59,r_{14}=61,\\ r_{15}=67,r_{16}=71,r_{17}=73,r_{18}=79,r_{19}=83,r_{20}=89,r_{21}=97,\\ r_{22}=101,r_{23}=103,r_{24}=107,r_{25}=109,r_{26}=113,r_{27}=121,\\ r_{28}=127,r_{29}=131,r_{30}=139,r_{31}=143,r_{32}=149,r_{33}=151,\\ r_{34}=157,r_{35}=163,r_{36}=167,r_{37}=169,r_{38}=173,r_{39}=179,\\ r_{40}=181,r_{41}=187,r_{42}=189,r_{43}=191,r_{44}=193,r_{45}=197,\\ r_{46}=199,r_{47}=209,r_{48}=211\}.\end{array}$

3.3. Factor Representation and eCTC Construction

Theorem 1

Let $\alpha =q_j{h}_{i/j}$ , where $q_j,h_{i/j}\in S_7$ and $\alpha$ has no factors of 2, 3, 5, and 7.

Then there exist values $k_{i/j}\in {\mathbb{N}}_0$ (with $0\le k_{i/j}\le 211$ ) and $t_{i/j}\in S_7$ such that

$\alpha =t_{i/j}+210k_{i/j}=q_j{h}_{i/j}$ (2)

This theorem enables the construction of eCTC₂₁₀, a $48\times 48$ table whose entries are expressed as:

$\left(k_{i/j},t_{i/j}\right)$ ,

satisfying Equation (2).

The table has:

• 48 columns, indexed by $j=1,\cdots ,48$ , corresponding to values of $q_j$ .

• 48 rows, indexed by $i=1,\cdots ,48$ , corresponding to values of $h_{i/j}$ .

A schematic for eCTC₂₁₀ is shown in Table 1. The full table is presented in Table 2(a) (for $j=1,\cdots ,23$ ) and Table 2(b) (for $j=24,\cdots ,48$ ).

3.4. Illustrative Entries in the eCTC

From Table 1:

• For $j=1$ to $23$ , the corresponding values are $q_1=11,\cdots ,q_{23}=107$ .

• For $j=24$ to $46$ , the values are $q_{24}=107,\cdots ,q_{46}=199$ .

• For $j=47,48$ , the values are $q_{47}=209$ and $q_{48}=211$ .

As examples:

• For $\left(i,j\right)=\left(1,1\right)$ ,

$k_{1/1}=\frac{11\times 11-t_{1/1}}{210}$ ,

given $k_{1/1}=0$ and $t_{1/1}=121$ .

Thus, the entry is $\left(k_{1/1},t_{1/1}\right)=\left(0,121\right)$ .

• For $\left(i,j\right)=\left(1,2\right)$ ,

$k_{1/2}=\frac{13\times 11-t_{1/2}}{210}$ ,

yielding $k_{1/2}=0$ and $t_{1/2}=143$ .

Thus, $\left(k_{1/2},t_{1/2}\right)=\left(0,143\right)$ .

• For $\left(i,j\right)=\left(48,48\right)$ ,

$k_{48/48}=\frac{211\times 211-t_{48/48}}{210}$ ,

giving $k_{48/48}=211$ and $t_{48/48}=211$ .

3.5. Mirror and Symmetry Properties

Inspection of eCTC₂₁₀ reveals striking mirror and symmetry phenomena among composites.

The relationships justify dividing eCTC₂₁₀ into areas A, B, C, D, E, each of which can be subdivided into four regions, such as A₁, A₂, A₃, A₄. Table 1(b) specifies the ranges for each region; for example, region A₁ corresponds to $1\le i\le 22$ and $1\le j\le 23$ .

3.6. Mirror Rules

Remark 1

Given a composite

Table 1. A schematic of eCTC₂₁₀: (a) Divided regions of eCTC₂₁₀; (b) Ranges of i and j for eCTC₂₁₀ regions.

(a)

(b)

${\alpha }_1=q_j\times h_{i/j}$ , $q_j,h_{i/j}\in S_7$ ,

we define

1) Vertical mirror:

${\alpha }_2=q_j\times \left(210-h_{i/j}\right)$ .

2) Horizontal mirror:

${\alpha }_3=\left(210-q_j\right)\times h_{i/j}$ .

3) Diagonal mirror:

${\alpha }_4=\left(210-q_j\right)\times \left(210-h_{i/j}\right)$ .

Theorem 2

Let $\left(k_{i/j},t_{i/j}\right)$ be an element in region A₁, and

${\alpha }_{a,1}=t_{i/j}+210k_{i/j}=q_j\times h_{i/j}$

is a composite.

Then, in Table 1(a), we find that the vertical, horizontal, and diagonal mirrors of ${\alpha }_{a,1}$ :

1) Vertical mirror (region A₂)

$\left(q_j-1-k_{i/j},210-t_{i/j}\right)$

corresponds to a composite number

${\alpha }_{a,2}=q_j\left(210-h_{i/j}\right)$ .

2) Horizontal mirror (region A₃)

$\left(h_{i/j}-k_{i/j}-1,210-t_{i/j}\right)$

corresponds to a composite number

${\alpha }_{a,3}=\left(210-q_j\right)\times h_{i/j}$ .

3) Diagonal mirror (region A₄)

$\left(210-q_j-h_{i/j}+k_{i/j},t_{i/j}\right)$

corresponds to a composite number

${\alpha }_{a,4}=\left(210-q_j\right)\left(210-h_{i/j}\right)$ .

The vertical, horizontal, and diagonal mirrors of the B, C, D, E regions are derived similarly.

Example 1

Take $q_1=11$ , and $\left(k_{1/1},t_{1/1}\right)=\left(0,121\right)$ , so that

${\alpha }_{a,1}=121$ .

Then,

1) Vertical mirror

$\left(10,89\right)\Rightarrow 11\times 89$

2) Horizontal mirror

$\left(10,89\right)\Rightarrow 199\times 11$

3) Diagonal mirror

$\left(188,121\right)\Rightarrow 199\times 199$

3.7. Consequences of Mirror and Symmetry Effects

The symmetry properties imply that:

• It suffices to generate entries $\left(k_{i/j},t_{i/j}\right)$ for the regions A₁, B₁, C₁, D₁, E₁.

• All remaining entries in regions A₂, A₃, A₄, B₂, B₃, B₄, C₂, C₃, C₄, D₂, D₃, D₄, E₂, E₃, and E₄ follow directly by symmetry.

Examining Table 2(a) and Table 2(b), we conclude:

• Composites without factors 2, 3, 5, and 7 appear in an unexpectedly orderly manner in the eCTC.

• Every such composite has exactly three mirror counterparts.

• The eCTC provides an efficient mechanism for listing all such composites in any interval

$\left[11+210k,211+210k\right]$ , $0\le k\le 210$ .

The detailed procedure is given in the next section.

4. Formula of Primes

Based on Statement 5, we attain the following results.

Theorem 3 (Cyclic Effect)

Given an element $\left(k_{i/j},t_{i/j}\right)$ in eCTC₂₁₀ with a corresponding composite

$\alpha =t_{i/j}+210k_{i/j}=q_j\times h_{i/j},$

where $h_{i/j}\in S_7$ , let $k^{\text{*}}\in N^{+}$ with $k^{\text{*}}\le 211$ , and suppose

$k^{\text{*}}=k_{i/j}+q_{j}m,m\in N^{+}$ .

Then,

${\alpha }^{\text{*}}=t_{i/j}+210k^{\text{*}}$

is also a composite with the same root $t_{i/j}$ and cycle value m.

Proof

We have

${\alpha }^{\text{*}}=t_{i/j}+210\left(k_{i/j}+q_{j}m\right)=q_j{h}_{i/j}+210q_{j}m=q_j\left(h_{i/j}+210m\right)$ . (3)

Hence, ${\alpha }^{\text{*}}$ is a composite with root value $t_{i/j}$ .

Remark 2 (Reduced Kin Value)

For fixed $q_j$ and $k^{\text{*}}$ , define

$k_j\left(k^{\text{*}}\right)=\text{min}\left\{k^{\text{*}}-q_{j}m|m\in {\text{N}}_0,0\le k^{\text{*}}-q_{j}m\le 211\right\}$ .

The meaning of $k_j\left(k^{\text{*}}\right)$ is: given $q_j\in S_7$ and an integer $k^{\text{*}}$ with $0\le k^{\text{*}}\le 210$ , there exists some m such that

$0\le k^{\text{*}}-q_{j}m\le q_j$ ,

and this value is taken as $k_j\left(k^{\text{*}}\right)$ . If $k^{\text{*}}=0$ , then $m=0$ and hence $k_j\left(0\right)=0$ .

Formula: Listing Composites in $\left[11+210k^{*},211+210k^{*}\right]$

Consider integers of the form

$\alpha =r_i+210k^{\text{*}},r_i\in S_7$ .

Such an $\alpha$ is a composite (without factors 2, 3, 5, 7) if and only if there exists an element $\left(k_{i/j},t_{i/j}\right)$ in eCTC₂₁₀ such that

$k_{i/j}=k_j\left(k^{\text{*}}\right)$ , $t_{i/j}=r_i$ .

Table 2. (a) A full eCTC₂₁₀ table, for $j=1,2,\cdots ,23$ ; (b) A full eCTC₂₁₀ table, for $j=24,25,\cdots ,48$ .

(a)

(b)

Proof

We have

$\alpha =r_i+210k^{\text{*}}=t_{i/j}+210k^{\text{*}}$ .

By definition of $k_j\left(k^{\text{*}}\right)$ of Remark 2, we can write

$k^{\text{*}}=k_{i/j}+q_{j}m$

for some $m\in N_0$ . Thus,

$\alpha =t_{i/j}+210k_{i/j}+210q_{j}m$ .

From Theorem 3, we know $t_{i/j}+210k_{i/j}=q_j{r^{\prime }}_i$ , for some ${r^{\prime }}_i\in S_7$ .

Hence,

$\alpha =q_j\left({r^{\prime }}_i+210m\right)$ ,

which shows that $\alpha$ is a composite number.

Example 2: Composites in $\left[\text{11, 211}\right]$

We first list composites in the base interval [11, 211]. Here,

$\alpha =r_i+210k^{\text{*}}$ , $k^{\text{*}}=0$ .

From eCTC₂₁₀ (Table 2(a) and Table 2(b)), when $k_{i/j}=k_j\left(0\right)=0$ , the entries with $k_{i/j}=0$ and values $t_{i/j}$ are:

$\left(0,t_{i/j}\right)\in \left\{\left(0,121\right),\left(0,143\right),\left(0,187\right),\left(0,143\right),\left(0,169\right),\left(0,187\right)\right\}$

Thus, the composites in [11, 211] of the form $\alpha =r_i+210\times 0$ are

$r_i\in \left\{121,143,169,187\right\}$ ,

as listed in Table 3(a).

Example 3: Composites in $\left[11+210,211+210\right]$

Now consider the interval $\left[11+210,211+210\right]=\left[221,421\right]$ . Here,

$\alpha =r_i+210k^{\text{*}}$ , $k^{\text{*}}=1$ .

Compute

$k_j\left(1\right)=\text{min}\left\{1-q_{j}m\right\}$

For all relevant $q_j$ , the minimum is attached at $m=0$ , so $k_j\left(1\right)=1$ .

The entries in eCTC₂₁₀ with $k_{i/j}=1$ include:

$\begin{array}{l}\left(1,t_{i/j}\right)\in \{\left(1,43\right),\left(1,109\right),\left(1,131\right),\left(1,197\right),\left(1,11\right),\cdots ,\left(1,193\right),\\ \left(1,11\right),\cdots ,\left(1,181\right),\left(1,37\right),\cdots ,\left(1,151\right)\}\end{array}$

Therefore, the composites in [221, 421] (without factors 2, 3, 5, 7) are

$\alpha =r_i+210$ ,

where $r_i$ belongs to the root set

${\text{RS}}_1=\left\{11,37,43,79,89,109,113,131,151,167,181,193,197\right\}$ ,

as shown in Table 3(b).

Table 3. Example of a composites listing.

(a)
$j$			$q_j$			$k_j\left(0\right)$			$h_{i/j}$
1			11			0			121				143			187
2			13			0			143				169
3			17			0			187
4			19			0
5			23			0
6			29			0
7			31			0
8			37			0
9			41			0
10			43			0
11			47			0
12			53			0
13			59			0
14			61			0
15			67			0
16			71			0
17			7			0
18			73			0
19			83			0
20			89			0
21			97			0
22			101			0
23			103			0
24			107			0
25			109			0
26			113			0
27			121			0
28			127			0
(b)
$j$	$q_j$			$k_j\left(1\right)$			$h_{i/j}$
1	11			1			43			109					131			197
2	13			1			11			37		89			167			193
3	17			1			11			79		113			181
4	19			1			37			113		151
5	23			1			43			89		189
6	29			1			109			167
7	31			1			131			193
8	37			1			197
9	41			1
10	43			1
11	47			1
12	53			1
13	59			1
14	61			1
15	67			1
16	71			1
17	7			1
18	73			1
19	83			1
20	89			1
21	97			1
22	101			1
23	103			1
24	107			1
25	109			1
26	113			1
27	121			1
28	127			1
(c)
$j$		$q_j$			$k_j\left(100\right)$			$h_{i/j}$
1		11			1			43			109			131			197
2		13			9			47			73			151
3		17			15			29			97			131			199
4		19			5			71			109
5		23			8			137
6		29			13			83			199
7		31			7			173
8		37			26			53			127
9		41			18			197
10		43			14			113			199
11		47			6			103
12		53			47			41
13		59			41			181
14		61			39			167
15		67			33			—
16		71			29			—
17		7			27			97
18		73			21			—
19		83
20		89
21		97
22		101			100			109
23		103
24		107			100			79
25		109			100			37
26		113			100			131
27		121
28		127			100			209

Example 4: Composites in $\left[11+210\times 100,211+210\times 100\right]$

Consider the interval

$\left[11+210\times 100,211+210\times 100\right]=\left[21011,21211\right]$

Here,

$\alpha =r_i+210\times 100$ , $k^{\text{*}}=100$

Compute $k_j\left(100\right)$ for various $q_j$ :

$k_1\left(100\right)=\text{min}\left\{100-11m\right\}=1$ ,

$k_2\left(100\right)=\text{min}\left\{100-13m\right\}=9$ ,

$k_3\left(100\right)=\text{min}\left\{100-17m\right\}=15$ ,

$k_4\left(100\right)=\text{min}\left\{100-19m\right\}=5$ ,

$k_5\left(100\right)=8$ , $k_6\left(100\right)=13$ , $k_7\left(100\right)=7,\cdots ,k_{28}\left(100\right)=209$ .

From eCTC₂₁₀ (Table 2(a) and Table 2(b)) we observe:

• For $k_1\left(100\right)=1$ , the related $h_{i/j}$ values are 43, 109, 131, 197.

• For $k_2\left(100\right)=9$ , the related $h_{i/j}$ values are 47, 73, 151, $\cdots$

• For $k_{15}\left(100\right)=33$ , there are no corresponding $h_{i/j}$ values.

• For $k_{28}\left(100\right)=100$ , the related $h_{i/j}$ is 209.

Hence, the composites in [21011, 21211] (without factors 2, 3, 5, 7) are

$\alpha =r_i+210\times 100$ ,

where $r_i$ belongs to the root set

$\begin{array}{l}{\text{RS}}_2=\{29,37,41,43,53,71,73,79,83,97,103,109,113,127,131,151,\\ 167,173,181,197,199,209\},\end{array}$

as listed in Table 3(c).

Remark 3 (Prime Listing via Composite Removal)

To list all primes in the interval $\left[11+210k^{\text{*}},211+210k^{\text{*}}\right]$ , we proceed as follows:

1) First, list all integers in the interval that are coprime to 2, 3, 5, 7, i.e., all$\alpha =r_i+210k^{\text{*}}$ with $r_i\in S_7$ .

2) Then, remove from this set all composites (without factors 2, 3, 5, 7) found by the Formula of listing composites above.

The remaining integers are precisely the primes in that interval.

Formula: Prime Listing in $\left[11+210k^{*},211+210k^{*}\right]$

Integers

$\alpha =r_i+210k^{\text{*}}$ , $r_i\in S_7$ ,

are prime if and only if they are not among the composites found by formula “Formula of listing composite within $\left[11+210k^{\text{*}},211+210k^{\text{*}}\right]$ .”

Example 5: Prime Numbers in Several Intervals

1) Primes in $\left[11,211\right]$

From Example 2, the composites are 121, 143, 169, 187. Hence, the primes are

$\alpha =r_i+210\times 0$ , $r_i\in S_7$ , $r_i\notin \left\{121,143,169,187\right\}$ .

2) Primes in $\left[221,421\right]$

From Example 3, the composites correspond to RS₁. Thus, the primes are

$\alpha =r_i+210$ , $r_i\in S_7$ , $r_i\notin {\text{RS}}_1$ ,

where RS₁ is given above.

3) Primes in $\left[21011,21211\right]$

From Example 4, the composites correspond to RS₂. Thus, the primes are

$\alpha =r_i+210\times 100$ , $r_i\in S_7$ , $r_i\notin {\text{RS}}_2$ .

Formula of Factorization

The table eCTC₂₁₀ can also be used to factor a composite.

Consider a composite $\alpha$ satisfying

$11\le \alpha \le {211}^2$ , $\gcd \left(\alpha ,210\right)=1$ .

Then $\alpha$ has a root $r_i\in S_7$ and period $k^{\text{*}}\le 210$ such that

$\alpha =r_i+210k^{\text{*}}$ .

Factorization Formula

The composite $\alpha$ contains a factor $q_j\in S_7$ if and only if there exists an element $\left(k_{i/j},t_{i/j}\right)$ in eCTC₂₁₀ such that

$k_{i/j}=k^{\text{*}}$ , $t_{i/j}=r_i$ .

Example 6: Factorizing 2431

We have

$2431=121+210\times 11$

and the corresponding $\left(k^{\text{*}},r_i\right)=\left(11,121\right)$ .

From eCTC₂₁₀ for $q_2=13$ , $q_3=17$ , we have

$\left(k_{40/2},t_{40/2}\right)=\left(11,121\right)$ , $\left(k_{31/3},t_{31/3}\right)=\left(11,121\right)$ .

Hence, 2431 contains factors 13 and 17. Also, $121=11\times 11$ . Therefore,

$2431=13\times 17\times 11$ .

Example 7: Factorizing 39203

We write:

$39203=143+210\times 186$ .

Thus, the root is $r_i=143$ and period $k^{\text{*}}=186$ .

From eCTC₂₁₀ at $q_j=197$ , we find the entry

$\left(186,143\right)$ ,

implying 197 divides 39203.

Therefore,

$39203=197\times 199$ .

5. Extension and Discussion

In this section, we extend the proposed framework to larger moduli and discuss related formulas, computational complexity, and potential applications.

5.1. Extension from S₇ to S₁₁, S₁₃, …

The eCTC and the prime formulas above are based on the set $S_7$ and the modulus

$2\times 3\times 5\times 7=210$ ,

so that the associated primes are limited to those smaller than 211². To study where larger primes hide, we extend the set from $S_7$ to $S_{11}$ , and further to $S_{13},S_{17},\cdots$

Let

${\lambda }_x=2\times 3\times 5\times \cdots \times x$ ,

and let $S_x$ be the set of positive integers up to ${\lambda }_x+1$ that are not divisible by any prime in $\left\{2,3,5,\cdots ,x\right\}$ . Although the size of $S_x$ increases with x, the mirror and symmetry effects observed in the eCTC for $S_7$ remain structurally the same for all $S_x$ .

Figure 1 conceptually explains why higher-level sequences $S_x$ share the same features as $S_7$ :

1) Growth of $\left|S_x\right|$

Denote $\left|S_x\right|$ as the number of elements in $S_x$ . Clearly, $\left|S_x\right|$ grows as $x$ increases, reflecting the finer filtration by more primes.

2) Complementary pairs

For each $S_x$ , certain pairs of elements maintain a complementary relationship.

Figure 1. Extension from sequence $S_7$ to higher sequences.

For example:

• in $S_7$ ,

$11+199=13+197=\cdots =103+107=210$ .

• in $S_{11}$ ,

$13+2297=17+2293=\cdots =1153+1157=2310$ .

• Even in $S_{23}$ ,

$29+223092841=\cdots =111546433+111546437=223092870$ .

Thus, complementary pairs always sum to ${\lambda }_x$ (or its analogue), preserving a balanced structure.

3) Final rows and doubling property

In each $S_x$ , the last two entries form a special pair whose sum equals twice the product $2\times 3\times 5\times \cdots \times x$ .

• In $S_7$ , the last two elements are 209 and 211, with

$209+211=2\times 210$ .

• In $S_{11}$ , the last two are 2309 and 2311, with

$2309+2311=2\times 2310$ .

4) Partition into regions

These structural properties ensure that the eCTC for $S_x$ can always be partitioned into regions $A_l$ , $B_l$ , $C_l$ , $D_l$ , $E_l$ , for $l=1,2,3,4$ , in a manner analogous to eCTC₂₁₀. Each eCTC contains a fixed, structured number of elements:

• In eCTC₂₁₀, there are $\left(5+6+1\right)\times 4=48$ elements.

• In eCTC₂₃₁₀, there are $\left(59+60+1\right)\times 4=480$ elements.

• In eCTC₃₀₀₃₀, there are $\left(719+720+1\right)\times 4=5760$ elements.

This extension allows us to locate larger primes and factor larger composites efficiently.

Example 8: Factoring a Semiprime 5,246,767

Consider the semiprime 5,246,767. Since

${210}^2<5246767<{2310}^2$ ,

we work with eCTC₂₃₁₀.

We write

$5246767=757+2310\times 2271$ .

From the eCTC for ${\lambda }_{11}=2310$ , we identify $r_{100}=757$ . For $j=100$ , we have $q_i=757$ and $q_j=2311$ .

Moreover,

$2271=757+757\times 2+2311\times 0$ .

Therefore,

$5246767=757\times \left(2311+2310\times 2\right)=757\times 6931$ .

5.2. Additional Prime Formulas

We now derive several additional formulas related to prime distribution, twin primes, Pythagorean primes, prime tuples, and neighboring primes.

Let

$d_i\left(k\right)=\{\begin{array}{l}1,\text{ if }\alpha =r_i+210k\text{ is prime},\\ 0,\text{ otherwise},\end{array}$

for $i=1,2,\cdots ,48$ and $0\le k\le 210$ .

Using the prime-listing formula, we obtain:

1) Formula for Prime Distribution

• the number of primes in the interval $\left[11+210k,211+210k\right]$ is

${\displaystyle \sum }_{i=1}^{48}{d}_i\left(k\right)$ .

• The total number of primes in a range of blocks, say from $k=k_1$ to $k=k_2$ , is

${\displaystyle \sum }_{k=k_1}^{k_2}{\displaystyle \sum }_{i=1}^{48}{d}_i\left(k\right)$

2) Formula for the n-th prime

Let the largest prime in the interval $\left[11+210k,211+210k\right]$ be the $n\text{-th}$ prime in the sequence of primes. Then

$n=4+{\displaystyle \sum }_{j=0}^k{\displaystyle \sum }_{i=1}^{48}{d}_i\left(j\right)$ ,

where the constant 4 accounts for the primes 2, 3, 5, 7, which are not included in the $S_7$ -based representation (with 2 regarded as the first prime).

3) Formula for Twin Primes

Primes $\alpha$ and ${\alpha }^{\prime }$ form a twin prime pair if

$\alpha =r_i+210k$ , ${\alpha }^{\prime }=r_{i+1}+210k$ , $r_{i+1}=r_i+2$ .

In other words, $\alpha$ and ${\alpha }^{\prime }$ differ by 2 and are twin primes.

4) Formula for Pythagorean Primes

A prime $\alpha$ is called a Pythagorean prime if $a\equiv 1$ (mod 4), i.e.,

$\alpha =4l+1$ , $l\in N^{+}$ ,

and it can be expressed as

$\alpha =m^2+n^2$ ,

where m is even, and n is odd.

A prime $\alpha =r_i+210k$ is Pythagorean if:

a) $r_i=4l+1$ . The kin value k is even,

b) $r_i=4l+3$ . The kin value k is odd,

for some $l\in N^{+}$ .

5) Formula for l-Tuple Primes

A sequence of primes $p_a$ , $p_b$ , $p_c$ , $\cdots$ , $p_h$ is called an l-tuple of primes if

$p_b-p_a=p_c-p_b=p_d-p_c=\cdots =p_h-p_g=l$ .

In our framework, a sequence of primes ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n$ forms an n-tuple of primes with step $210l$ and root $r_i$ (base 210) if:

${\alpha }_1=r_i+210k$

${\alpha }_2=r_i+210\left(k+l\right)$

${\alpha }_3=r_i+210\left(k+2l\right)$

$⋮$

${\alpha }_n=r_i+210\left(k+\left(n-1\right)l\right)$

6) Formula for Neighboring Primes

A sequence of primes ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n$ are called neighboring primes if there exist $r_i$ and k such that

${\alpha }_1=r_i+210k$ , ${\alpha }_2=r_{i+1}+210k$ , ${\alpha }_3=r_{i+2}+210k$ , $\cdots$ , ${\alpha }_n=r_{i+n-1}+210k$ .

Example 9: 30-Tuple Primes in [120,000,000, 122,000,000]

We seek primes forming 30-tuples in the interval [120,000,000, 122,000,000].

Since

$\sqrt{122000000}\approx 11045$ ,

we choose $S_{13}$ with modulus

$2\times 3\times 5\times 7\times 11\times 13=30030$ .

We consider a 30-tuple of the form:

$p_a=r_i+30030k$ ,

$p_b=r_i+30+30030k$ ,

$p_c=r_i+60+30030k$ ,

$p_d=r_i+90+30030k$ ,

$p_e=r_i+120+30030k$ ,

$p_f=r_i+150+30030k$ .

Solving within the target interval yields

$k=4035$ , $r_i=3761$ ,

and the corresponding primes in a 30-tuple are:

$p_a=121174811$ ,

$p_b=121174841$ ,

$p_c=121174871$ ,

$p_d=121174901$ ,

$p_e=121174931$ ,

$p_f=121174961$ .

6. Conclusion

In this study, we derive the kin-root representation for composite numbers without factors 2, 3, 5, and 7. We construct an extended Cyclic Table of Composites using kin-root tuples, and reveal mirror and symmetry properties within the table. We then attain formulas of primes. Primes within a given interval can be located by eliminating all composites related to the table. Additional examples and extensions to larger cases are provided.

Funding

This work is supported in part by CityU 9610556.

Acknowledgements

We are grateful for the in-kind support from NTU and for the constructive comments on the article from the anonymous reviewers.

Conflicts of Interest

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

References

[1]	Dickson, L.E. (2005) History of the Theory of Numbers, Volume II: Diophantine Analysis. Dover Publications.
[2]	Zagier, D. (1977) The First 50 Million Prime Numbers. The Mathematical Intelligencer, 1, 7-19.[CrossRef]
[3]	Stein, M.L., Ulam, S.M. and Wells, M.B. (1964) A Visual Display of Some Properties of the Distribution of Primes. The American Mathematical Monthly, 71, 516-520.[CrossRef]
[4]	Ribenboim, P. (1996) The New Book of Prime Number Records. Springer.[CrossRef]
[5]	Crandall, R. and Pomerance, C. (2005) Prime Numbers: A Computational Perspective. Springer.
[6]	Aiyar, V.R. (1934) Sundaram’s Sieve for Prime Numbers. The Mathematics Student, 2, 73.
[7]	Atkin, A.O.L. and Bernstein, D.J. (2003) Prime Sieves Using Binary Quadratic Forms. Mathematics of Computation, 73, 1023-1030.[CrossRef]
[8]	Pritchard, P. (1982) Explaining the Wheel Sieve. Acta Informatica, 17, 477-485.[CrossRef]
[9]	Kumanduri, R. and Romero, C. (1998) Number Theory with Computer Applications. Prentice Hall.
[10]	Matiyasevich, Y.V. (1999) Formulas for Prime Numbers. In: Tabachnikov, S., Ed., Kvant Selecta: Algebra and Analysis (Vol. 2), American Mathematical Society, 13-24.
[11]	Zagier, D. (1997) Newman’s Short Proof of the Prime Number Theorem. The American Mathematical Monthly, 104, 705-708.[CrossRef]
[12]	Li, H., Fang, S., Lin, B.M.T. and Kuo, W. (2023) Unifying Colors by Primes. Light: Science & Applications, 12, Article No. 32.[CrossRef] [PubMed]
[13]	Li, H., Huang, Y., Fang, S. and Kuo, W. (2021) A Prime-Logarithmic Method for Optimal Reliability Design. IEEE Transactions on Reliability, 70, 146-162.[CrossRef]
[14]	Li, H., Fang, S. and Kuo, W. (2024) The Periodic Table of Primes. Advances in Pure Mathematics, 14, 394-419.[CrossRef]
[15]	Li, H., Fang, S., Kuo, W. and Lin, N. (2025) Listing Prime Numbers Periodically. Advances in Pure Mathematics, 15, 247-268.[CrossRef]
[16]	Li, H., Fang, S., Kuo, W. and Lin, N. (2025) Kernel Factor Pairs for Semiprime Factorization. Advances in Pure Mathematics, 15, 629-642.[CrossRef]