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.

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 × 3 × 5 × 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].

Statement1

Let

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:

Label these integers as

Statement2

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

where r i ∈ S 7 is called the root, k is an integer (the kin), 210 is the period, and 0 ≤ k ≤ 210 . Thus, all such integers fall into 48 congruence classes modulo 210.

Statement3

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

where q j ∈ S 7 divides α and m is a positive integer.

Statement4

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

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

Statement5

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

and consider

Then, α ′ has a kin value that is q j cycles larger than that of α , and satisfies

Thus, composites repeat periodically at intervals proportional to their factor q j .

Statement6

Each composite

has three corresponding “mirror” composites:

Vertical mirror:

Horizontal mirror:

Diagonal mirror:

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

Statement7

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

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

Statement8

With the eCTC, we derive:

a formula for locating primes efficiently within an interval,a factorization formula based on kin-root relationships, anda tuple-prime formula for identifying prime pairs and higher-order prime clusters.

Statement9

Although Statements 1-8 are based on λ 7 = 210 , the method extends naturally to larger moduli. For example:

One may construct:

a kin-root table eCTC₂₃₁0 to find primes in

or a kin-root table eCTC₃₀₀₃₀ to find primes in

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

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

Let

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

Based on Statement 5, we attain the following results.

Theorem3 (Cyclic Effect)

Given an element ( k i / j , t i / j ) in eCTC₂₁₀ with a corresponding composite

where h i / j ∈ S 7 , let k * ∈ N + with k * ≤ 211 , and suppose

Then,

is also a composite with the same root t i / j and cycle value m.

Proof

We have

Hence, α * is a composite with root value t i / j .

Remark 2 (Reduced Kin Value)

For fixed q j and k * , define

The meaning of k j ( k * ) is: given q j ∈ S 7 and an integer k * with 0 ≤ k * ≤ 210 , there exists some m such that

and this value is taken as k j ( k * ) . If k * = 0 , then m = 0 and hence k j ( 0 ) = 0 .

Formula: Listing Composites in [ 11 + 210 k * ,   211 + 210 k * ]

Consider integers of the form

Such an α is a composite (without factors 2, 3, 5, 7) if and only if there exists an element ( k i / j , t i / j ) in eCTC₂₁₀ such that

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

(a)

(b)

Proof

We have

By definition of k j ( k * ) of Remark 2, we can write

for some m ∈ N 0 . Thus,

From Theorem 3, we know t i / j + 210 k i / j = q j r ′ i , for some r ′ i ∈ S 7 .

Hence,

which shows that α is a composite number.

Example 2: Composites in [ 11, 211 ]

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

From eCTC₂₁₀ (Table 2(a) and Table2(b)), when k i / j = k j ( 0 ) = 0 , the entries with k i / j = 0 and values t i / j are:

Thus, the composites in [11, 211] of the form α = r i + 210 × 0 are

as listed in Table 3(a).

Example 3: Composites in [ 11 + 210 , 211 + 210 ]

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

Compute

For all relevant q j , the minimum is attached at m = 0 , so k j ( 1 ) = 1 .

The entries in eCTC₂₁₀ with k i / j = 1 include:

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

where r i belongs to the root set

as shown in Table 3(b).

Table 3. Example of a composites listing.

Example 4: Composites in [ 11 + 210 × 100 , 211 + 210 × 100 ]

Consider the interval

Here,

Compute k j ( 100 ) for various q j :

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

For k 1 ( 100 ) = 1 , the related h i / j values are 43, 109, 131, 197.For k 2 ( 100 ) = 9 , the related h i / j values are 47, 73, 151, ⋯ For k 15 ( 100 ) = 33 , there are no corresponding h i / j values.For k 28 ( 100 ) = 100 , the related h i / j is 209.

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

where r i belongs to the root set

as listed in Table 3(c).

Remark 3 (Prime Listing via Composite Removal)

To list all primes in the interval [ 11 + 210 k * , 211 + 210 k * ] , we proceed as follows:

1) First, list all integers in the interval that are coprime to 2, 3, 5, 7, i.e., all α = r i + 210 k * with r i ∈ 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 [ 11 + 210 k * , 211 + 210 k * ]

Integers

are prime if and only if they are not among the composites found by formula “Formula of listing composite within [ 11 + 210 k * , 211 + 210 k * ] .”

Example 5: Prime Numbers in Several Intervals

1) Primes in [ 11 , 211 ]

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

2) Primes in [ 221 , 421 ]

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

where RS₁ is given above.

3) Primes in [ 21011 , 21211 ]

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

Formula of Factorization

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

Consider a composite α satisfying

Then α has a root r i ∈ S 7 and period k * ≤ 210 such that

Factorization Formula

The composite α contains a factor q j ∈ S 7 if and only if there exists an element ( k i / j , t i / j ) in eCTC₂₁₀ such that

Example 6: Factorizing 2431

We have

and the corresponding ( k * , r i ) = ( 11 , 121 ) .

From eCTC₂₁₀ for q 2 = 13 , q 3 = 17 , we have

Hence, 2431 contains factors 13 and 17. Also, 121 = 11 × 11 . Therefore,

Example 7: Factorizing 39203

We write:

Thus, the root is r i = 143 and period k * = 186 .

From eCTC₂₁₀ at q j = 197 , we find the entry

implying 197 divides 39203.

Therefore,

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

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.

This work is supported in part by CityU 9610556.

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