1
The Periodic Table of Primes
Han-Lin Li1, Shu-Cherng Fang2, Way Kuo3*
1Department of Computer Science, City University of Hong Kong
2Department of Industrial and Systems Engineering, North Carolina State University
3Hong Kong Institute for Advanced Study, City University of Hong Kong
*Corresponding author: Way Kuo, way@cityu.edu.hk
Abstract
Prime numbers are roots of integers1. Over millennia, nobody has been able to predict where
prime numbers sprout2 or how they spread. It has been believed that primes grow like weeds
among natural numbers. This study nevertheless establishes the Periodic Table of Primes
(PTP) using four prime numbers of 2, 3, 5, and 7. We identify 48 integers out of a period of
2 × 3 × 5 × 7 = 210 to be the roots of all primes and composites without factors of 2, 3, 5, and
7, each of which is an offspring of the 48 integers uniquely allocated on the PTP. The PTP
provides a platform to make the study of primes clearer and easier. In other words, the 48
integers serve as the genes of primes, and among them 15 pairs serve as the genes of twin
primes.
This study identifies the cyclic effects among the composites, spawning primes distributed
periodically. Three major contributions made in the article are the Formula of Primes, the
Periodic Table of Primes, and the Counting Functions for Primes and Twin Primes. These
findings provide answers to questions of interest to the academic community, such as finding
a future prime, factoring an integer, illustrating the fundamental theorem of arithmetic,
predicting the total number of primes and twin primes or estimating the maximum prime gap
within an interval, among others.
Introduction
The fundamental theorem of arithmetic is one of the greatest theorems of mathematics1,
which states that a composite integer can be represented uniquely as a product of primes. In
recent years, primes as roots of integers have been studied and widely applied to data
science1, cryptography2, and engineering design3, etc.
For a long time, many said these integer roots grew like weeds among natural numbers, but
nobody could predict where the next primes may sprout4. Many believe that primes are
unpredictable. Oliver and Soundaranjan5 investigated the distribution of consecutive primes,
while Luque and Lacasa6 reported patterns of the last digit of primes. Wang7 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 prime listing, prime
factorization, counting functions for primes and twin primes, or the next prime of a known
number. Studies published to predict primality and pattern of primes are ad hoc with
tremendous limitations and uncertainties.
Some6 observed that primes located near each other tend to avoid repeating their last digits,
which indicates that primes are not distributed as random as theorists often assume. Others
such as Tóth8 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.
In addition to the work by Zhang9, Maynard10 reported the minimum upper bound for prime
gaps. Also reported includes applications of the Maynard-Tao sieve method11. Few spells out
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the locations where a maximum prime gap in an interval will appear. Furthermore, there is no
study or intent on visualizing the fundamental theorem of arithmetic to illustrate the spatial
relationship between a composite number and its factors.
Initialized by Gauss, there were studies on counting the total numbers of primes and twin
primes within an interval, but only approximations were obtained 12,13. 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.14 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(gray) are represented by 3 × 5 = 15, 2 × 5 = 10, 2 × 3 = 6, and 2 × 3 × 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 14.
Inspired by C235, we intend to:
•
•
•
find a set of integers which serves as the root of primes and twin primes,
establish the Formula of Primes and build the Periodic Table of Primes (PTP) that
allocate primes, and
form the Counting Functions for Primes and Twin Primes.
Also accomplished include to predict within an interval, the largest k-tuple primes8 with the
same difference as well as to find the conditions for a maximum prime gap to occur, and to
visualize the fundamental theorem of arithmetic.
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) identifying the locations of 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 × 3 × 5 × 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 r1 = 11, r2 = 13, r3 = 17, ⋯ , r23 =
103, r24 = 107, ⋯ , r48 = 211. These r𝑖 s are placed on the left column of a table.
2. Developing the Cyclic Table of Composites (CTC)
The CTC consists of multiple 48 × 48 matrices [𝑙(i, j)] derived from the positions of
composites without factors of 2, 3, 5, and 7.
Within an interval of [48(θ − 1) + 1, 48θ], where θ ∈ N+ is considered a cycle, there are
48 integers which do not contain factors of 2, 3, 5, and 7. These 48 × θ integers are either
primes or composites, denoted as q1 = r1 = 11, q2 = 𝑟2 = 13, q3 = r3 = 17, ⋯ , q48 =
r48 = 211, for cycle 1; q49 = q1 + 210 = 221, q50 = q2 + 210 = 223, ⋯ , q96 = q48 + 210 =
421, for cycle 2; … ; q48(θ−1)+1 = q48(θ−2)+1 + 210 = q1 + 210(θ − 1), q48(θ−1)+2 =
q48(θ−2)+2 + 210 = q2 + 210(θ − 1), ⋯ , and q48θ = q48(θ−1) + 210 = q48 + 210(θ − 1), for
cycle θ . We call q1 to q48 segments of cycle 1, q49 to q96 segments of cycle 2, …, and
q48×(θ−1)+1 to q48θ segments of cycle θ. The qj s 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 [𝑙(i, j)], where index i refers to ri and j refers to qj , such that in each row of this
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table, pairs of entries share the same value of elements 𝑙(i, j)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 47th row and the 48th row of CTC(1). We define the basic CTC
(CTC(basic)) to be a 24 × 48 matrix of [𝑙(i, j)], i = 1, 2, ⋯ , 23, 47 and j = 1, 2, ⋯ , 48.
Utilizing the mirror effect, we find the complementary CTC (CTC(complementary)) a
24 × 48 matrix made of [𝑙(i, j)] for i = 24, 25, ⋯ , 46, 48 and j = 1, 2, ⋯ , 48. Combining the
CTC(basic) and the CTC(complementary), we form CTC(1).
We observe further an intermedia effect in CTC(1) and a cyclic effect between different
cycles of composite tables where each cycle is a 48 × 48 matrix. The entries of a higher
cycle are self-reproductive of CTC(1). By utilizing the intermedia and the cyclic effects, we
generate the CTC with multiple cycles, CTC(1), CTC(2), CTC(3), ⋯, CTC(θ).
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 α ∈
[0, b] without factors of 2, 3, 5, and 7 is a prime if and only if there exists an i ∈
{1, 2, … , 48} such that α = ri + 210k, for some integer k with k + 1 ∉
L𝑏 (i), where L𝑏 (i) is a set of 𝑙(i, j)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′) as a periodic table of primes from periods k 0 to k′, k 0 < k′ ∈ N+ .
PTP(k 0 , k′) is a 48 × (k ′ − k 0 + 1) matrix composed of all primes within an interval [a, b],
for a = 11 + 210 × k 0 and b = 211 + 210 × k′. Denote PTP + (k 0 , k′) as a matrix composed
of primes in PTP(k 0 , k′) plus composites without factors 2, 3, 5, and 7 within the interval
[a, b].
The steps of building a PTP are depicted in Figure 1. Starting from the CTC(initial), we
form a CTC(initial dual). By combing 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 readily available. Utilizing CTCs, we generate PTP+(k 0 , k′), where
we delete all composites to obtain PTP(k 0 , k′). The red and green points in Figure 1
represent the diagonal and intermediate effects, respectively. The red and green points
also show the cyclical pattern on CTC and the periodical pattern on PTP+.
Based on the CTCs and the PTPs, we form the Counting Functions to predict the total
number of primes and twin-prime pairs within an interval, instead of just an
approximations13.
Methods 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 unless stated otherwise.
In CTC, consisting of 48 rows and 48 columns, denote r𝑖 as the ith row and qj the jth column.
Let
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S1 =
=
S2 =
=
{r1 , r2 , … , r23 , r47 }
{11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 209},
{r24 , r25 , … , r46 , r48 }
{107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193,
197, 199, 211}, and
S = S1 ∪ S2 = {r1 , r2 , … , r48 } ={q1 , q2 , … , q48 }.
Note that r𝑖 s and qj s are integers in [11, 211] without factors of 2, 3, 5, and 7. Let (i, j) be at a
position in the CTC and 𝑙(i, j) be the corresponding entry of point (i, j), where 1 ≤ i, j ≤ 48,
determined by
𝑙(i, j) = 1 +
qj ×qh(i,j) −r𝑖
210
,
⋯ Expression (1)
where qh(i,j) ∈ S is a minimum integer such that qj × q𝐡(𝐢,𝐣) − r𝑖 is a multiplier of 210.
Expression (1) states that a composite β, without factors of 2, 3, 5 and 7, generated by β =
qj × q𝐡(𝐢,𝐣) , q𝐡(𝐢,𝐣) being one of {q1 , q2 , … , q48 }, is the ((𝑙(i, j) − 1)th descent of ri. If 𝑙(i, j) = 1, then
β = r𝑖 and β is the root ri itself. In general, if 𝑙(i, j) = n, then β = r𝑖 + 210 × (n − 1) is the nth
descendant of root r𝑖 . All CTC(θ)s are similarly specified.
An outline of CTC is shown in Figure 2. Various effects in CTC are described by the following
statements. Presented in Table S1(a) along with an illustrative example are given in the
Supplement under Example for Table S1(a). Observing Table S1(a), note that 𝑙(i, i) = qi +
1, 𝑖 = 1, 2, … , 23, 47. We identify the diagonal effect in CTC in Statement 1 below.
Statement 1 (The Diagonal effect of the CTC)
In the CTC, 𝒍(𝐣, 𝐣) = 𝐪𝐣 + 𝟏, for 𝐣 = 𝟏, 𝟐, ⋯ , 𝟒𝟖. Moreover, 𝒍(𝐣, 𝐣) ≥ 𝒍(𝐢, 𝐣), for 𝐢, 𝐣 =
𝟏, 𝟐, ⋯ , 𝟒𝟖 𝐚𝐧𝐝 𝐢 ≠ 𝐣.
Analyzing the distribution of 𝑙(i, j) in the ith row of Table S1(a), we find the dual effect between
𝑙(i, j) and 𝑙(i, h), described in Statement 2 below.
Statement 2 (The Dual effect of the CTC)
In the CTC, for any 𝒍(𝐢, 𝐣), 𝐢, 𝐣 = 𝟏, 𝟐, ⋯ , 𝟒𝟖 𝐚𝐧𝐝 𝐢 ≠ 𝐣, there exists an 𝐡(𝐢, 𝐣) ∈ {𝟏, 𝟐, ⋯ , 𝟒𝟖}
such that 𝒍(𝐢, 𝐣) = 𝒍(𝐢, 𝐡(𝐢, 𝐣)).
Statement 2 implies that given 𝑙(i, j), we know 𝑙(I, h(i, j)), under which situation j and h(i, j) are
dual to each other. Take 𝑖 = 1 and r1 = 11 for instance, 𝑙(1, 5) = 𝑙(1, 8) = 5, where q5 × q8 =
23 × 37 = 851 = 11 + 210(5 − 1). Notice that when h(i, j) = j, qj 2 is a descendant of r𝑖 and
𝑙(i, j) appears in the ith row only once. For instance, 𝑙(48,9) = 8 appears only once and 412 =
1682 = 211 + 210× (8 − 1).
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 Tables S1(a) and S1(b), we
also identify the mirror effect between each row of the CTC(basic) and its complementary row
in CTC deduced below.
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Statement 3 (The Mirror effect in the CTC)
The mirror effect in the CTC exists between pairs of all rows such that
(i)
𝒍(𝟒𝟕 − 𝐢, 𝐣) + 𝒍(𝐢, 𝐣) = {
(ii)
𝒍(𝟒𝟕, 𝐣) + 𝒍(𝟒𝟖, 𝐣) = {
𝐪𝐣 + 𝟏, 𝐢𝐟 𝐢 ≠ 𝐣
for 𝐢 = 𝟏, 𝟐, ⋯ 𝟐𝟑, 𝐣 = 𝟏, ⋯ , 𝟒𝟖.
𝟐𝐪𝐣 + 𝟏, 𝐢𝐟 𝐢 = 𝐣,
𝐪𝐣 , 𝐢𝐟 𝐣 = 𝟏, ⋯ , 𝟒𝟔
𝟐𝐪𝐣 , 𝐢𝐟 𝐣 = 𝟒𝟕, 𝟒𝟖
Statement 3 implies that if we know 𝑙(i, j) for 1 ≤ i ≤ 23, then we can find 𝑙(47 − i, j). Similarly,
if we know 𝑙(47, j), then we find 𝑙(48, j). Following Statement 3, we form the
CTC(complementary), which is the complementary of the CTC(basic).
CTC(complementary) is presented as Table 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 1.
Consider CTC in Figure 2. There are two red diagonals shown in Figure 2, which further
elaborates Table S1(a) and Table S1(b), The first diagonal is 𝑙(1, 1) = 12, 𝑙(2, 2) = 14, ⋯ ,
and 𝑙(46, 46) = 200, and the second one is 𝑙(46, 1) = 11, 𝑙(45, 2) = 13, ⋯, and 𝑙(1, 46) = 199.
The two cross lines in Figure 2 are obvious due to the mirror effect. 𝑙(47, j) and 𝑙(48, j), j =
1, 2, 3, ⋯ , 48, are mirror with each other. An Example is given in the Supplement under An
Associated Example for Statement 3.
Table S1(a) and Table S1(b) show another distribution of 𝑙(i, j) 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 to show how to transfer CTC to PTP. In Figure 2, the
red and green numbers represent the diagonal and the intermediate effects, respectively.
Expanding Expression (1) by denoting 𝑙(i, j + 48(θ − 1)) as the entries of CTC(θ), where θ ∈
N+ , we find the cyclic effect between CTC(1) and CTC(θ) made in Statement 4. Part of the
upper and lower rows of CTC(2) are illustrated in Table S2(a) and Table S2(b), respectively.
Statement 4 (The Cyclic effect between CTC(1) and CTC(𝛉))
The cyclic effect exists between CTC(1) and CTC(𝛉), 𝛉 = 2, 3, …, such that
𝒍(𝐢, 𝐣 + 𝟒𝟖(𝛉 − 𝟏)) = 𝒍(𝐢, 𝐣) + (𝛉 − 𝟏)𝐪𝐡(𝐢,𝐣) , 𝐟𝐨𝐫 𝟏 ≤ 𝐢, 𝐣 ≤ 𝟒𝟖, where
𝒍(𝐢, 𝐣) = 𝟏 +
𝐪𝐣 ×𝐪𝐡(𝐢,𝐣) −𝐫𝒊
𝟐𝟏𝟎
is the (i, j) entry in CTC(1).
Through 𝑙(i, j), ri , qj at cycle 1 (i.e., θ = 1), we find 𝑙(i, j + 48(θ − 1)) for cycle θ. An example is
given in the Supplement under Numerical Illustrations for Statement 4.
Proofs for the four above Statements are given in the Supplement under Proofs of the
Statements. With the dual, the diagonal, the mirror, the intermedia, and the cyclic effects
based on the 48 roots, composites without factors of 2, 3, 5, and 7 are distributed periodically
and cyclically.
Results
This study provides three major results: the Formula of Primes, the Periodic Table of Primes,
and the Counting Functions for Primes and Twin Primes.
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Given a positive integer b with b – 211 being a multiplier  of 210, and i ∈ {1, 2, ⋯ , 48}, define
L𝑏 (i) ≡ {𝑙(i, j + 48(θ − 1)|j = 1, 2, ⋯ , 48, and θ = 1, 2, … ,  .
(2)
⋯Expression
Specifically, L𝑏 (i) is a set consisting of 𝑙(i, j) ≤ ∆i,θ+1 , for j = 1, 2, …, 48θ, where
∆𝑖,θ = min{𝑙(i, j)|j = 48(θ − 1) + 1, 48(θ − 1) + 2, ⋯ , 48θ, for θ ∈ N+ }, and 211∆i,θ+1 ≥ b .
j
Notice that 𝑙(i, j + 48(θ − 1)) = 𝑙(i, j) + (θ − 1)q𝐡(𝐢,𝐣) where q𝐡(𝐢,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 r𝑖 ∈ S such that α − r𝑖 is a multiple of 210. Moreover, if α is not a composite number, then
it is a prime. From Expression (2) and Statement 4, we know that for any composite β ≤ b,
which does not contain factors of 2, 3, 5, and 7, it can be expressed by
β = qj × q𝐡(𝐢,𝐣) = ri + 210[𝑙(i, j) + (θ − 1)q𝐡(𝐢,𝐣) − 1] = ri + 210[𝑙(i, h(i, j)) + (θ − 1)qj − 1],
where 𝑙(i, j) + (θ − 1)q𝐡(i,𝐣) and 𝑙(i, h(i, j)) + (θ − 1)qj ∈ Lb (i).
We summarize the Formula of Primes below:
The Formula of Primes
An integer 𝛂 ∈ [𝟏𝟏, 𝐛] containing no factors of 2, 3, 5, and 7 is a prime if and only if
there
exists 𝐫𝐢 ∈ 𝐒 and 𝒌 ∈ N0 𝐬𝐮𝐜𝐡 that 𝛂 = 𝐫𝐢 + 𝟐𝟏𝟎𝐤, with k + 1 ∉ 𝐋𝐛 (𝐢), 𝐢 = 𝟏, 𝟐, … , 𝟒𝟖.
An integer α = ri + 210k ≤ b is a prime, if there does not exist a non-negative integer m which
satisfies k + 1 = 𝑙(i, j) + m × q𝐡(𝐢,𝐣) , for i = 1, 2, …, 48. A list of L𝑏 (1) for various b up to 44521
is listed in Table S3, which is presented in the Supplement under Justification for Establishing
Lb (1) and Table S3.
According to CTC (θ) and the Formula of Primes, we specify the Periodic Table of Primes:
The Periodic Table of Primes from period 𝐤 𝟎 𝐭𝐨 𝐩𝐞𝐫𝐢𝐨𝐝 𝐤 ′
For given 𝐤 𝐨 ∈ 𝐍𝟎 , 𝐤 ′ ∈ 𝐍+ , 𝐚𝐧𝐝 𝐤 ′ > 𝐤 𝐨 , 𝐏TP (𝐤 𝟎 , 𝐤 ′ ) is a 𝟒𝟖 × (𝐤 ′ − 𝐤 𝟎 + 𝟏) matrix
[𝛌(𝐢, 𝐤 + 𝟐𝟏𝟎)], where 𝛌(∙) is an integer such that
𝟏𝟏 + 𝟐𝟏𝟏 × 𝐤 𝟎 ≤ 𝛌(𝐢, 𝐤 + 𝟐𝟏𝟎) = 𝐫𝒊 + 𝟐𝟏𝟎𝐤 ≤ 𝟐𝟏𝟏(𝐤 ′ + 𝟏),
for k ∈ 𝐍𝟎 𝐰𝐢𝐭𝐡 𝐤 + 𝟏 ∉ 𝐋𝐛 (𝐢) 𝐚𝐧𝐝 𝐛 = 𝟐𝟏𝟏 (𝐤 ′ + 𝟏).
We denote PTP + (k o , k ′ ) for a matrix composed of primes in PTP(k o , k ′ ) plus composites
without factors of 2, 3, 5 and 7, within an interval [11 + 211 × k o , 211(k ′ + 1)].
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 [11, 1 + 2102 ] converted from Figure 2. Figure 3
has 48 rows and 211 columns, for i = 1, 2, ⋯ , 48, and k = 0, 1, 2, 3, ⋯ , 210. λ(i, k) is an integer
of the entry (i, k), computed from 𝑙(i, j) of Figure 2. The two dotted red curves in Figure 3 are
configured from two diagonal red lines of Figures 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.
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Table 1 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 Gauss12,13, the number of primes no more than b is
approximately
b
ln b
. We find further the exact Counting Function for Primes π(b) when b - 211
is a multiplier of 210:
The Counting Function for Primes
𝟒𝟖
𝛑(𝐛) = 𝟒 + 𝟒𝟖 × (𝐤 + 𝟏) − ∑|𝐋𝐛 (𝐢)|.
⋯ 𝐄𝐱𝐩𝐫𝐞𝐬𝐬𝐢𝐨𝐧 (𝟑)
𝒊=𝟏
Currently, few predict the number of twin-prime pairs in an interval [2, b]. For b – 211 as a
multiplier of 210, denote T(i, i + 1) as the number of twin-prime pairs on PTP(0, ⌈
b
211
⌉), i.e.,
T(i, i + 1) is a set of k with k + 1 ∉ Lb (i) and k + 1 ∉ Lb (i + 1), where i and i + 1 are for the ith
and the (i + 1)st row on the PTP, respectively. Our study finds the exact Counting Function
for Twin Primes π∗ (b) below:
The Counting Function for Twin Primes
𝛑∗ (𝐛) = 𝟑 + ∑|𝐓(𝐢, 𝐢 + 𝟏)|,
𝐢∈𝐖
⋯ 𝐄𝐱𝐩𝐫𝐞𝐬𝐬𝐢𝐨𝐧 (𝟒)
where 𝐖 = {𝟏, 𝟑, 𝟔, 𝟗, 𝟏𝟑, 𝟏𝟔, 𝟐𝟐, 𝟐𝟒, 𝟑𝟎, 𝟑𝟑, 𝟑𝟕, 𝟒𝟎, 𝟒𝟑, 𝟒𝟓, 𝟒𝟕}.
Proofs of Expressions (3) and (4) are given in the Supplement under Proofs of Expressions
(3) and (4), and examples of computing π(b) and π∗ (b) are illustrated in the Supplement
under Nine Examples for the Predictions of Primes by the PTP.
Discussion
The unpredictability of prime numbers forms a basis of many applications, one being
encryption called the RSA algorithm 2. However, neither the ancient Sieve of Eratosthenes’ 15
nor the modern Sieve of Atkin’s15 algorithms have ever elaborated the physical meaning of
finding primes. Torquato, Zhang and Courcy-Ireland16 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 primesrelated 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. Concluded below are some further thoughts.
1. Instead of choosing 2, 3, 5 and 7, one may choose 2, 3, 5, 7, and 11 as the base. By so
doing, the PTP will become gigantic, too complicated, and difficult to visualize, although
Electronic copy available at: https://ssrn.com/abstract=4742238
8
likely more effective. If one is interested in the behaviors of super-large primes, then such
a large prime table could be useful.
2. The PTP is helpful in explaining many unclear phenomena. For instance, it explains a
troublesome observation6 from the past which 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 3 or 7 or 9. From the
Formula of Primes, if a given prime is 221 (i. e. , θ = 2, r𝑖 = 11), then the most possible
near primes should be firstly 223 (i. e. , θ = 2, r𝑖 = 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 descendent 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 to be 1, 3, 7,
and 9 when primes go to infinity.
4. After the work of Zhang9, Maynard10 claimed that most existing resources used in finding
the infinitely achievable small gaps between primes adopted his approach. Based on the
CTC and the PTP, we readily find the maximum prime gap for various intervals. This
argument is amended under Justification of Claim 4 in Discussion as well as depicted in
Figure S1(a) and Figure S1(b) of the Supplement.
5. Examples for the predictions of primes are given in the Supplement under Nine Examples
for the Predictions of Primes by the PTP. Figures S1 and S2 in the Supplement are
depicted for visualizing the fundamental theorem of arithmetic and identifying the
maximum prime gap or the largest twin-prime pair in a specific interval, respectively.
References
1. Cartesian orthogonal coordinate system. Encyclopedia of Mathematics.
https://encyclopediaofmath.org/wiki/Cartesian_orthogonal_coordinate_system (2014).
2. Dickon, L. E., History of the Theory of Numbers, Volume II: Diophantine Analysis.
Dover Publications, Inc. (2005).
3. Li, H. L., Huang, Y. H., Fang, S. C. & Kuo, W., A prime-logarithmic method for optimal
reliability design. IEEE Trans. Reliability, 70, 146-162 (2021).
4. Zagier, D. The first 50 million prime numbers. The Mathematical Intelligencer 1 (Suppl
1), 7–19 (1977). https://doi.org/10.1007/BF03039306
https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BF03039306/fulltext.pdf
5. Oliver, R. J. L., & Soundaranjan, K., Unexpected biases in the distribution of
consecutive primes, Proc of the National Academy of Sciences, 113, E4446E4454. (2016). https://doi.org/10.1073/pnas.1605366113
6. Luque, B. & Lacasa, L., The first-digit frequencies of prime numbers and Riemann
Zeta zeros. Proceedings of the Royal Society A, 564, 2197-2216 (2009).
https://doi.org/10.1098/rspa.2009.0126
7. Wang, X. The genesis of prime numbers—revealing the underlying periodicity of
prime numbers. Advances in Pure Mathematics, 11, 12-18 (2021).
DOI: 10.4236/apm.2021.111002.
8. Tóth, L., “On the asymptotic density of prime k-tuples and a conjecture of Hardy and
Littlewood” Computational Methods in Science and Technology, 25(3) (2019).
9. Zhang, Y., "Bounded gaps between primes". Ann. of Math. 179 (3): 1121–1174
(2014).
10. Maynard, J. Small gaps between primes. Ann. Math. 181(2), 383–413 (2015).
11. Freiberg T., A note on the Theorem of Maynard and Tao, in Advances in the Theory
of Numbers, Springer, (2015), https://link.springer.com/chapter/10.1007/978-1-49393201-6_4
Electronic copy available at: https://ssrn.com/abstract=4742238
9
12. Zagier, D., Newman’s short proof of the prime number theorem, American Math
Monthly, 104(8), 705-708 (1997).JSTOR 2975232.
13. Nicomachus of Gerasa, Introduction to Arithmetic, 2 nd century.
14. Li, H. L., Fang, S. C., Lin, B. M. T. & Kuo, W., Unifying colors by primes. Light Sci
Appl. 12, 32 (2023). https://doi.org/10.1038/s41377-023-01073-x
15. Atkin, A. O. L. & Bernstein, D. J., Prime sieves using binary quadratic forms, Math.
Comp. 73, 1023-1030 (2004).
16. Torquato, S., Zhang, G. & Courcy-Ireland, M. de, Uncovering multiscale order in the
prime numbers via scattering, J. of Statistical Mechanics: Theory and Experiment
(2018).
Electronic copy available at: https://ssrn.com/abstract=4742238
CTC(initial)
𝑙(𝑖
r𝑖
, 𝑗) j
1 2
q1 q2
⋯
23
q23
1 r1
2 r2
⋮
seg
root i
dual effect
r𝑖
j
24 25
q24 q25
⋮
⋮
⋮
CTC(complementary)
j 1 2
me
𝑙(𝑖 nt qj
,𝑗
q1 q2
)
root i ri
seg
1 r1
2 r2
⋯ 23 24
q23 q24
⋯
46 47 48
q46 q47 q48
seg
mirror
effect
me
j
nt
𝑙(𝑖 qj
,𝑗
root i ri )
24 r24
25 r25
⋮
⋮
1 2 ⋯ 23 24
q1 q2
q23 q24
⋯
46 47 48
q46 q47 q48
⋮
23 r23
46 r46
47 r47
48 r48
CTC(1)
CTC(2)
seg
1 2 ⋯ 23 24 ⋯ 46 47 48 49 50 ⋯ 71 72 ⋯ 94 95 96
me
nt
q
𝑙(𝑖
q23 q24 ⋯ q46 q47 q48 q49 q50 q71 q72 ⋯ q94 q95 q96
root i r , 𝑗) q1 q2
1 r1
2 r2
CTC(θ)
48(θ-1)+1
q48(θ-1)+1
j
𝑗
⋮ ⋮
23 r23
24 r24
⋮
10
46
q46
23 r23
CTC(basic)
𝑖
⋯
1 r1
2 r2
23 r23
⋮
me j
nt
q
)
root i
me j
nt
q
,𝑗
𝑙(𝑖
seg
CTC(initial dual)
48θ-2 48θ-1 48θ
q48θ-2 q48θ-1 q48θ
cyclic
effect
⋮
47 r47
48 r48
PTP+(0, 210)
period k
1 2
𝜆
(𝑖,k
)
root i r𝑖
1 r1
2 r2
⋯
PTP+(211, 420)
210 211 212
⋯
420
PTP+(210(θ-1)+1, 210θ)
⋯
210(θ-1)+1
⋮ ⋮
23 r23
24 r24
⋮
⋮
47 r47
48 r48
Figure 1: Schematic process of forming the PTP
Note:
for diagonal effect.
for intermediate effect.
Electronic copy available at: https://ssrn.com/abstract=4742238
θ≥2
210θ
seg 𝑗
me
nt
𝑙(𝑖 q𝑗
,
root i r𝑖 𝑗)
Electronic copy available at: https://ssrn.com/abstract=4742238
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101103107109113121127131137139143149151157163167169173179181187191193197199 209 211
1 11 12 2 2 9 5 16 7 5 7 29 24 33 40 39 37 52 59 57 63 9 67 88 24 84 16 37 111 33 39 29 40 67 57 52 63 84 59 145155145 88 155111182182199 199 12
2 13 10 14 13 7 12 24 25 36 23 38 41 34 31 57 7 29 43 60 29 59 79 12 75 31 98 23 43 85 34 59 85 131 13 75 60 25 24 79 10 41 38 98 131 57 197 36 197 14
3 17
18
74 32
193
193 18
4 19
20
22 86
191
191 20
5 23
24
21 87
187
187 24
6 29
30
10 71 35 100
181
181 30
7 31
32
19 89
179
179 32
8 37
38
9 69 37 101
173
173 38
9 41
42
68 38
169
169 42
10 43
44
16 92
167
167 44
163
163 48
11 47
48
15 93
12 53
54
7 65 41 103
157
157 54
13 59
60
12 96
151
151 60
14 61
62
6 63 43 104
149
149 62
15 67
68
10 98
143
143 68
16 71
72
9 99
139
139 72
17 73
74
60 46
137
137 74
18 79
80
7 101
131
131 80
CTC(initial)
CTC(initial dual)
19 83
84
6 102
127
127 84
20 89
90
56 50
121
121 90
21 97
98
54 52
113
113 98
22 101
102 53 53 109
109 102
23 103
104107
107 104
24 107
103108
103 108
25 109
101 51 55 110
101 110
26 113
97
50 56
114
97 114
27 121
89
48 58
122
89 122
83
98 6
128
83 128
28 127
29 131
79
97 7
132
79 132
30 137
73
44 62
138
73 138
31 139
71
95 9
140
71 140
32 143
67
94 10
144
67 144
33 149
61
96 41 65 6
150
61 150
34 151
59
92 12
152
59 152
35 157
53
95 39 67 7
158
53 158
36 163
47
89 15
164
47 164
37 167
43
88 16
168
43 168
38 169
41
36 70
170
41 170
39 173
37
93 35 71 9
174
37 174
40 179
31
85 19
180
31 180
41 181
29
92 33 73 10
182
29 182
42 187
23
83 21
188
23 188
43 191
19
82 22
192
19 192
44 193
17
30 76
194
17 194
45 197
13
90 29 77 12
198
13 198
46 199 11
80 24
200 11 200
47 209 1 7 3 1 8 25 22 3 33 17 15 26 34 52 15 47 8 38 17 64 91 38 26 80 68 7 34 101 68 122 47 111 22 64 80 111101 33 159 52 25 122181159 91 181 210 210
48 211 10 6 14 18 15 4 9 34 8 26 32 27 25 9 52 24 65 41 66 25 6 63 77 27 41 106 87 26 63 15 92 32 127 87 77 52 66 136 14 127156 65 10 34 106 18 208 212
Note: Numbers in red — for diagonal effect. Numbers in green — for intermediate effect.
Figure 2: An outline of CTC(1)
11
12
Electronic copy available at: https://ssrn.com/abstract=4742238
𝑖
period k
1
2
3
⋮
𝜆(𝑖,
r𝑖
k)
23
48
72
96
110
120
11
13
17
21
22
23
24
25
26
⋮
97
101
103
107
109
113
44
45
46
47
48
193
197
199
209
211
Note:
0
for diagonal effect.
for intermediate effect.
Figure 3: An outline of PTP+(0, 210)
130
140
150
160
170
180
190
210
Table 1: A Partial table of PTP+[0, 48)
period k
es/co
mpo
sites
r𝑖
prim
root i
Electronic copy available at: https://ssrn.com/abstract=4742238
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101
103
107
109
113
11x11
127
131
137
139
11x13
149
151
157
163
167
13x13
173
179
181
11x17
191
193
197
199
11x19
211
0
1
2
3
4
Note: Numbers in red — for composites
13
5
6
7
8
9
10
11
431 641 23x37 1061 31x41 1481 19x89 1901 2111 11x211
223 433 643 853 1063 19x67 1483 1693 11x173 2113 23x101
227 19x23 647 857 11x97 1277 1487 1697 1907 29x73
229 439 11x59 859 1069 1279 1489 1699 23x83 2129
233 443 653 863 29x37 1283 1493 13x131 1913 11x193
239 449 659 11x79 13x83 1289 1499 1709 19x101 2129
241 11x41 661 13x67 23x47 1291 19x79 29x59 17x113 2131
13x19 457 23x29 877 1087 1297 11x137 17x101 41x47 2137
251 461 11x61 881 1091 1301 1511 1721 1931 2141
11x23 463 673 883 1093 1303 1523 1723 1933 2143
257 467 677 887 1097 1307 37x41 11x157 13x149 19x113
263 11x43 683 19x47 1103 13x101 1523 1733 26x67 2153
269 479 13x53 29x31 1109 1319 11x139 37x47 1949 17x127 23x103
271 13x37 691 17x53 11x101 1321 1531 1741 1951 2161
277 487 17x41 907 1117 1327 29x53 1747 19x103 11x197
281 491 701 911 19x59 11x121 23x67 17x103 37x53 13x167
283 17x29 19x37 11x83 1123 31x43 1543 1753 13x151 41x53
17x17 499 709 919 1129 13x103 1549 1759 11x179 2179
293 503 23x31 13x71 11x103 17x79 1553 41x43 1973 37x59
13x23 509 719 929 17x67 19x71 1559 29x61 1979 11x199
307 23x37 727 937 31x37 23x59 1567 1777 1987 13x169
311 521 17x43 941 1151 1361 1571 13x137 11x181 31x71
313 523 733 23x41 1153 29x47 11x143 1783 1993 2203
317 17x31 11x67 947 13x89 1367 19x83 1787 1997 2207
11x29 23x23 739 13x73 19x61 37x37 1579 1789 1999 47x47
17x19 13x41 743 953 1163 1373 1583 11x163 2003 2213
331 541 751 31x31 1171 1381 37x43 1801 2011 2221
337 547 757 967 11x107 19x73 1597 13x139 2017 17x131
11x31 19x29 761 971 1181 13x107 1601 1811 23x47 23x97
347 557 13x59 977 1187 11x67 1607 23x79 2027 2237
349 13x43 769 11x89 29x41 1399 1609 17x107 2029 2239
353 563 773 983 1193 23x61 1613 1823 19x107 2243
359 569 19x41 23x43 11x109 1409 1619 31x59 2039 13x173
19x19 571 11x71 991 1201 17x83 1621 1831 13x157 2251 23x107
367 577 787 997 17x71 13x109 1627 11x167 23x89 37x61
373 11x53 13x61 17x59 1213 1423 23x71 19x97 2053 31x73
13x29 587 797 19x53 1217 1427 1637 1847 11x187 2267
13x13 379 19x31 17x47 1009 23x53 1429 11x149 43x43 29x71 2269
173 383 593 11x73 1013 1223 1433 31x53 17x109 2063 2273
179 389 599 809 1019 1229 1439 17x97 11x169 2069 43x53
181 17x23 601 811 1021 1231 11x131 13x127 1861 19x109 2281
11x17 397 607 19x43 13x79 1237 1447 1657 1867 31x67 2287
191 401 13x47 821 1031 17x73 1451 11x151 1871 2081 29x79
193 13x31 613 823 1033 11x113 1453 1663 1873 2083 2293
197 11x37 617 827 17x61 29x43 31x47 1667 1877 2087 2297 23x109
199 409 619 829 1039 1249 1459 1669 1879 2089 19x121 11x209
11x19 419 17x37 839 1049 1259 13x113 23x73 1889 2099 2309
211 421 631 29x29 1051 13x97 1471 41x41 31x61 11x191 2311
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101
103
107
109
113
121
127
131
137
139
143
149
151
157
163
167
12
13
14
15
16
17
18
19
20
21
13x17
22
23
48
47x103
41x241
9883
9887
13x211
17x211
29x211
43x103
11x29x31
23x211
41x103
13x761
19x521
9901
9907
37x103
53x187
31x103
23x431
29x103
47x211
9923
9929
9931
19x523
9941
61x163
9949
37x269
23x433
9967
13x13x59
9973
11x907
17x587
67x149
97x103
13x769
73x137
10007
10009
17x19x31
43x233
11x911
37x271
29x107
79x127
10037
10039
31x107
37x107
11x11x83
13x773
41x107
19x23x23
23x209
19x209
43x107
17x109
13x209
47x107
89x113
10061
29x347
10067
10069
10079
17x593
14
Supporting Information for
The Periodic Table of Primes
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) matrix for the
first cycle. In the CTC(initial), its entries of [𝑙(i, j)], for i = 1, 2, … , 23 and
j = 1, 2, … , 23, are both starting from 11; namely, the left column contains r1,
r2, …, r23 and the top row contains q1, q2, …, q23.
b. Applying the dual effect, we form another 23 × 23 CTC(initial dual) matrix
from CTC(initial). In this new matrix, its entries of [𝑙(i, j)], for
i = 1, 2, … , 23 and j = 24, 25, … , 46, are the first 23 roots starting from 11 for
“i”s and the 24th to 46th roots for “j”s, respectively. Namely, the left column
contains r1, r2, …, r23, and the top row contains q24, q25, …, q46.
c. Add r47 as the end of the 24th row and add q47 and q48 as the end of the 47th
and the 48th 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 matrices 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).
CTC (initial) = {𝑙(i, j) | i = 1, 2, ⋯ , 23; j = 1, 2, ⋯ , 23},
CTC (initial-dual) = {𝑙(i, j) | i = 1, 2, ⋯ , 23; j = 24, ⋯ , 46},
CTC(basic) = {𝑙(i, j) | i = 1, 2, ⋯ , 23, 47; j = 1, 2, ⋯ , 48},
CTC(complementary) = {𝑙(i, j) | i = 24, 25, ⋯ , 46, 48; j = 1, 2, ⋯ , 48},
CTC(1) = {𝑙(i, j) | i, j = 1, 2, ⋯ , 48}.
2. Example for Table S1(a)
Given q1 = 11, compute 𝑙(1, 1), as follows:
q ×q
𝐡(𝐢,𝐣)
∵ 𝑙(1, 1) = 1 + 1 210
q 𝐡(𝐢,𝐣) = 211,
∴ 𝑙(1, 1) = 1 +
−r1
11×211−11
210
, and the unique q 𝐡(𝐢,𝐣) to let
11×q𝐡(𝐢,𝐣) −11
210
be an integer is
= 1 + 11 = 12.
Similarly, we find
𝑙(2, 1), 𝑙(3, 1) 𝑙(4, 1), 𝑙(5, 1), 𝑙(6, 1), ⋯ , 𝑙(23, 1), 𝑙(47, 1) = (10, 6, 4, 11, 5, ⋯ , 8, 1).
3. Elaboration of the Dual Effect of the CTC(basic)
Electronic copy available at: https://ssrn.com/abstract=4742238
15
In the CTC(1), the dual effect made in Statement 2 can be further elaborated below
between 𝒍(𝐢, 𝐣) and 𝒍(𝐢, 𝐡(𝐢, 𝐣)).
(i) For 𝒍(𝐢, 𝐣) = 𝒍(𝐢, 𝐡(𝐢, 𝐣)), 𝐢𝐟 𝟏 ≤ 𝐣, 𝐡(𝐢, 𝐣) ≤ 𝟐𝟑, 𝐭𝐡𝐞𝐧
𝒍(𝐢, 𝟒𝟕 − 𝐣) = 𝒍(𝐢, 𝟒𝟕 − 𝐡(𝐢, 𝐣))
𝐪𝟒𝟕−𝐣 × 𝐪𝟒𝟕−𝐡(𝐢,𝐣) 𝐫𝒊
𝐫𝒊
=𝟏+
(
+ 𝒍(𝐢, 𝐣) − 𝟏) −
.
𝐪𝐣 × 𝐪𝐡(𝐢,𝐣)
𝟐𝟏𝟎
𝟐𝟏𝟎
(ii) For 𝒍(𝐢, 𝐣) = 𝒍(𝐢, 𝐡(𝐢, 𝐣)), 𝐟𝐨𝐫 𝟏 ≤ 𝐣 ≤ 𝟐𝟑 𝐛𝐮𝐭 𝐧𝐨𝐭 𝐟𝐨𝐫 𝐡(𝐢, 𝐣), then 𝟐𝟒 ≤ 𝐡(𝐢, 𝐣) ≤
𝐫 +𝟐𝟏𝟎[𝒍(𝐢,𝐣)−𝟏]
𝟒𝟔 𝐚𝐧𝐝 𝐪𝐡(𝐢,𝐣) = 𝒊
𝐪𝐣
.
By utilizing Statement 2, we can generate a new matrix CTC(initial dual) in the
Introduction from CTC(initial). Taking the first row, i.e. i = 1 and r1 = 11 as an
example (Figure 2):
(i)
Since 𝑙(1, 2) = 𝑙(1, 3) = 2, we have
𝑙(1, 47 − 2) = 𝑙(1, 47 − 3) = 1 +
q 45 × q 44 11
11
(
+ 𝑙(1, 2) − 1) −
= 182
q 2 × q 3 210
210
Similarly, since 𝑙(1, 5) = 𝑙(1, 8) = 5, we have
q 42 × q 39 11
11
𝑙(1, 47 − 5) = 𝑙(1, 47 − 8) = 1 +
(
+ 𝑙(1, 5) − 1) −
= 155
q 5 × q 8 210
210
(ii) For i = 1 and j = 1, 6, 10, 12, ⋯ , 19, 21, 22, we do not have 1 ≤ h(i, j) ≤ 23 to
fit
𝑙(1, j) = 𝑙(1, h(i, j)), then we compute 𝑙(1, h(i, j)) for 24 ≤ h(i, j) ≤ 46.
When j = 1, q1 = 11 and 𝑙(1, 1) = 12,
since
r1 +210(𝑙(1,1)−1)
q1
=
11+210(12−1)
11
= 211 = q 48 , we have
h(1, 1) = 48 and 𝑙(1, 48) = 𝑙(1, 1) = 12
When j = 6, q 6 = 29 and 𝑙(1, 6) = 16,
r +210(𝑙(1,6)−1)
since 1
q6
= 109 = q 25 , we have
h(1, 6) =25 and 𝑙(1, 25) = 𝑙(1, 6) = 16.
Similarly, we can predict 𝑙(1, 26) = 𝑙(1, 15) = 37, 𝑙(1, 28) = 𝑙(1, 12) =
33, 𝑙(1, 29) = 𝑙(1, 14) = 39, 𝑙(1, 30) = 𝑙(1, 10) = 29, ⋯ , 𝑙(1, 41) = 𝑙(1, 22) = 88.
4. An Associated Example for Statement 3
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Comparing the CTC(basic) with the CTC(complementary) and referring to Table
S1(a) and Table S1(b), it is clear that:
Given j = 1 and q1 = 11, we have 𝑙(1, 1) + 𝑙(46, 1) = 12 + 11 = 2q1 + 1.
Given j = 2 and q2 = 13, we have 𝑙(2, 2) + 𝑙(45, 2) = 14 + 13 = 2q 2 + 1.
Given j = 1 and q1 = 11, for i = 2, we have 𝑙(45, 1) + 𝑙(2, 1) = 2 + 10 = 12 = q1 +
1.
Given j = 12 and q12 = 53, for i = 1, 𝑙(46, 12) + 𝑙(1, 12) = 21 + 33 = 54 = q12 + 1;
for i = 23, 𝑙(24, 12) + 𝑙(23, 12) = 28 + 26 = 54 = q12 + 1;
for i=47, 𝑙(47, 12) + 𝑙(48, 12) = 26 + 27 = 53 = q12.
5. Statement and proof of the intermedia effect in the CTC(1)
Table S1(a) and Table S1(b) show another distribution of 𝑙(i, j) on the CTC, named
the intermedia effect, which happens at some symmetrical columns, described
below.
The intermedia effect in the CTC(1)
In CTC(1), there exists the intermedia effect between columns 23 and 24 such that,
for 𝐢 = 𝟏, 𝟐, … , 𝟒𝟖,
(i)
(ii)
if 𝒍(𝐢, 𝟐𝟑) + 𝒍(𝐢, 𝟐𝟒) = 𝟏𝟎𝟖, 𝐭𝐡𝐞𝐧 𝒍(𝐢, 𝟐𝟑) = 𝒍(𝟒𝟕 − 𝐢, 𝟐𝟒),
if 𝒍(𝐢, 𝟐𝟑) + 𝒍(𝐢, 𝟐𝟒) = 𝟏𝟎𝟔, 𝐭𝐡𝐞𝐧 𝒍(𝐢, 𝟐𝟑) + 𝟐 = 𝒍(𝟒𝟕 − 𝐢, 𝟐𝟒).
For instances, in the case of green columns 23 and 24 of Figure 2, 𝑙(19, 23) =
𝑙(28, 24) = 6, ⋯ , 𝑙(2, 23) + 2 = 𝑙(45, 24) = 77. Other column pairs also exhibit the
intermedia effect. For example, in the case of columns 22 and 25, if 𝑙(i, 22) +
𝑙(i, 25) = 110, then 𝑙(i, 22) = 𝑙(47 − i, 25).
Proof of the intermedia effect in the CTC
For the case of columns 23 and 24,
(i)
From the mirror effect, we have 𝑙(i, 24) + 𝑙(47 − i, 24) = q 24 + 1 = 108.
If 𝑙(i, 23) + 𝑙(i, 24) = 108, then 𝑙(i, 23) = 108 − 𝑙(i, 24) = 𝑙(47 − i, 24).
(ii)
If 𝑙(i, 23) + 𝑙(i, 24) = 106, then 𝑙(i, 23) + 2 = 108 − 𝑙(i, 24) = 𝑙(47 − i, 24).
For the case of columns 22 and 25, from the mirror effect, we know
𝑙(i, 25) + 𝑙(47 − i, 25) = q 25 + 1 = 109 + 1 = 110
If 𝑙(i, 22) + 𝑙(i, 25) = 110, then 𝑙(i, 22) = 𝑙(47 − i, 25).
6. Numerical Illustrations for Statement 4
Take θ = 2 (i.e., cycle 2) as an example. CTC(2) consisting of CTC(basic) and
CTC(complementary) is presented in Table S2(a) and Table S2(b), respectively.
Table S2(a) is for 𝑙(i, j + 48) where 1 ≤ i ≤ 23. Table S2(b) is for 𝑙(i, j + 48), where
24 ≤ i ≤ 46 and 47, 48.
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Table S2(a) can be deducted from the CTC(basic) of Table S1(a) by the cyclic effect
of Statement 4. For instance, 𝑙(i, 49) in Table S2(a), for 1 ≤ i ≤ 23, is computed as
follows
𝑙(i, 49) = 𝑙(i, 1) + (2 − 1)q 𝐡(𝐢,𝐣) = 𝑙(i, 1) +
r𝑖 + 210(𝑙(i, j) − 1)
q1
Since (𝑙(1,1), 𝑙(2,1), 𝑙(3,1), 𝑙(4,1), ⋯ , 𝑙(23,1)) = (12, 10, 6, 4, ⋯ , 8), and q1 = 11, we
have (𝑙(1,49), 𝑙(2,49), 𝑙(3,49), ⋯ , 𝑙(23,49)) = (223, 183, 103, 63, ⋯, 151).
Similarly, since (𝑙(1,2), 𝑙(2,2), 𝑙(3,2), 𝑙(4,2), ⋯ , 𝑙(23,2)) = (2, 14, 12, 11, ⋯ , 8), and
q1 = 11, we then have (𝑙(1,50), 𝑙(2,50), 𝑙(3,50), 𝑙(4,50), ⋯ , 𝑙(23,50)) =
(19, 225, 191, 174, ⋯ , 129).
Table S2(b) is deduced from Table S2(a) by the mirror effect of Statement 3. For
instance, for q 49 = 221, we have
𝑙(24, 49) = q 49 + 1 − 𝑙(23, 49) = 221 + 1 − 151 = 71, and
𝑙(46, 49) = 2q 49 + 1 − 𝑙(1, 49) = 442 + 1 −223 = 220.
Therefore, using the CTC(1), we can generate the CTC(θ), for θ = 2, 3, …
For instance, given r11 = 11, we have 𝑙(1,1) = l +
211, and 𝑙(1, 1) = 12.
11×q𝐡(𝐢,𝐣) −11
210
. Therefore, q 𝐡(𝐢,𝐣) =
For θ = 2, we have 𝑙(1, 49) = 12 + (2 − 1) × 211 = 223.
For θ = 3, we have 𝑙(1, 97) = 12 + (3 − 1) × 211 = 434.
For θ = 4, we have 𝑙(1, 145) = 12 + (4 − 1) × 211 = 645.
Similarly, 𝑙(1, 2) = 2. Therefore, we find 𝑙(1, 49) = 2 + (2 − 1) × 211 = 213 and
𝑙(1, 98) = 2 + (3 − 1) × 211 = 424.
7. Proofs of the Statements
Proof of Statement 1
For each i, j = 1, 2, ⋯ 48, notice that in CTC(1),
𝑙(j, j) = 1 +
q j × q 𝐡(𝐣,𝐣) − rj
210
=1+
q j (211 − 1)
210
= 1 + q j , as rj = q j and we choose q 𝐡(𝐣,𝐣) = q 48 = 211.
For a given j, ri = q j̃ × q 𝐡(𝐢,𝐣) − (𝑙(i, j) − 1) × 210, for some h(i, j). This means that a
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fixed ri is the remainder of q j̃ × q h(i,j) subtracting the maximum multiple of 210
smaller than q j × q 𝐡(𝐢,𝐣) . Since q h(j,j) = q 48 = 211 is the largest q 𝐡(𝐢,𝐣) hence 𝑙(j, j) ≥
𝑙(i, j).
Proof of Statement 2
By definition, 𝑙(i, j) = 1 +
qj qh(i,j) − ri
210
q
q − ri
, 𝑙(i, h(i, j)) = 1 + 𝐡(𝐢,𝐣)210𝐣
.
q q𝐡(𝐢,𝐣) − ri
Therefore, we have 𝑙(i, j) = 𝑙(i, h(i, j)) = 1 + j× 210
Proof of Statement 3
(i): Notice that r47−i + ri = 210, ∀ i = 1, ⋯ , 23,
q q𝐡(𝐢,𝐣) −r47−i
𝑙(47 − i, j) = 1 + j
𝑙(i, j) = 1 +
210
qj q𝐡′(𝐢,𝐣) −ri
210
for some q 𝐡′(𝐢,𝐣) and
for some q 𝐡′(𝐢,𝐣) .
Hence,
q j (q 𝐡(𝐢,𝐣) + q 𝐡′(𝐢,𝐣) ) − 210
210
q j (q 𝐡(𝐢,𝐣) + q 𝐡′(𝐢,𝐣) )
=1+
210
𝑙(47 − i, j)
+ 𝑙(i, j)
Since
=2+
qj (q𝐡(𝐢,𝐣) +q𝐡′(𝐢,𝐣) )
210
must be an integer, and q j has no factors of 2, 3, 5, 7,
q 𝐡′(𝐢,𝐣) + q 𝐡(𝐢,𝐣) must be a multiple of 210. This happens only when
q 𝐡(𝐢,𝐣) + q 𝐡′(𝐢,𝐣) = {
210, if h(i, j) + h′ (i, j) = 47
420, if h(i, j) or h′ (i, j) = 48
Therefore,
𝑙(47 − i, j) + 𝑙(i, j) = {
(ii):
q j + 1, if i ≠ j
2q j + 1, if i = j
Notice that r47 + r48 = 209 + 211 = 420. Hence
q q𝐡(𝐢,𝐣) −r47
j
𝑙(47, j) + 𝑙(48, j) = (1 +
210
=
q q𝐡′(𝐢,𝐣) −r48
) + (1 + j
210
)
qj (q𝐡(𝐢,𝐣) +q𝐡′(𝐢,𝐣) )
210
q j , if h(i, j) + h′ (i, j) = 47
={
2q j , if h(i, j) or h′(i, j) = 48
The 2nd case happens only when j = 47, 48.
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Proof of Statement 4
𝑙(i, j) + (θ − 1)q 𝐡(𝐢,𝐣) = 1 +
q j q 𝐡(𝐢,𝐣) − ri + 210(θ − 1)q 𝐡(𝐢,𝐣)
210
[q j + 210(θ − 1)]q 𝐡(𝐢,𝐣) − ri
210
q j+48(θ−1) q 𝐡(𝐢,𝐣) − ri
=1+
210
=1+
= 𝑙(i, j + 48(θ − 1))
8. Justification for Establishing 𝐋𝐛 (𝟏) and Table S3
Starting with θ = 1, compute q j , q h(1,j) and 𝑙(1, j) using Expression (1)
q j × q 𝐡(𝐢,𝐣) − 11
𝑙(1, j) = 1 +
, for j = 1, 2, ⋯ , 48.
210
Then by the cyclic effect of Statement 4, 𝑙(1, j + 48(θ − 1))s, for 2 ≤ θ ≤ 19, are
computed based on 𝑙(1, j):
𝑙(1, j + 48) = 𝑙(1, j) + 𝑞𝐡(𝟏,𝐣)
𝑙(1, j + 96) = 𝑙(1, j) + 2𝑞𝐡(𝟏,𝐣)
⋮
𝑙(1, j + 864) = 𝑙(1, j) + 18𝑞𝐡(𝟏,𝐣)
We then list 𝑙(i, j + 48(θ − 1)) for θ = 1, 2, 3, … , 19 in Table S3. The bottom row of
Table 3S shows that ∆1,1 = 2, ∆1,2 = 15, ∆1,3 = 28, ⋯ , ∆1,19 = 210. Notably,
min{𝑙(i, j + 48(θ − 1))} < min{𝑙(i, j + 48θ)} for all θ = 1, 2, 3, … , 19, and ∆i,1 < ∆i,2 <
j
j
∆i,3 < ⋯ , for all i.
For instance, given b = 2,311 = 211 + 210 × 10, compute L2,311 (1) by applying
Expression (2).
Referring to Expression (2), Lb (i) is a set consisting of 𝑙(i, j)  ∆i,θ+1. Since ∆𝑖,θ ≥
2311
⌈ 211 ⌉ = 11, choose θ = 2 to satisfy ∆i,θ = min{𝑙(1, j + 48)} = 15. We then obtain
j
L2311 (1) = {2, 5, 7, 9, 12, 15}.
According to the Formula of Primes, in order for α = 11 + 210k ≤ 2311 to be
primes, it requires k + 1 ∉ L2311 (1) for k ∈ 1, 2, … , 10. Therefore, we let k ∈
{0, 2, 3, 5, 7, 9, 10} and get the following primes: 11, 11 + 210 × 2 = 431, 11 +
210 × 3 = 641, 11 + 210 × 5 = 1061, 11 + 210 × 7 = 1481, 11 + 210 × 9 =
1901 and 11 + 210 × 10 = 2111. They are all shown in the first row of Table 1.
Given b = 211 + 210 × 28 = 6091, we choose θ = 3 to satisfy
∆i,θ = min{𝑙(1, j + 96)} = 28, then L6091 (1) =
j
{2, 5, 7, 9, 12, 15, 16, 17, 19, 23, 24, 28}.
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In order for α = 11 + 210k ≤ 6091 to be primes, we let k ∈
{0, 2, 3, 5, 7, 9, 10, 12, 13, 17, 19, 20, 21, 24, 25, 26} such that k+1∈ L6091 (1) to obtain
the following primes: 11, 431, 641, 1,061, 1,481, 1,901, 2,111 2,531, 2,741, 3,581,
4,001, 4,211, 4,421, 5,051, 5,261, 5,471.
Similarly, we can list all Lb (i) and all primes α = 11 + 210k ≤ b for a given b.
9. Proofs of Expression (3) and Expression (4)
Proof of Expression (3)
The total number of elements in PTP + (0, k) is 48 × (k + 1), for k ∈ N0 . Within the
interval [11, 2112 θ], the number of different 𝑙(i, j) is |Lb (i)| for b = 211 + 210k.
Hence, the total number of composites without 2, 3, 5, and 7 within the above
interval is |Lb (i)|. Note that 4 primes are less than 10. Therefore, the number of
primes within [2, b] is π(b) = 4 + 48 × (k + 1) − ∑48
𝑖=1|Lb (i)|.
Proof of Expression (4)
For an integer i ∈ W = {1, 3, 6, 9, 13, ⋯ , 45, 47}, (r𝑖 , r𝑖+1 ) are twin primes for r𝑖 ≥
11. If there exists a k, k ∈ 𝑁0, such that k + 1 ∉ Lb (i) and k + 1 ∉ Lb (i + 1), then
r𝑖 + 210k and r𝑖+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 [2, b] is π∗ (b) = 3 + ∑𝑖∈W|T(i, i + 1)|, where T(i, i + 1) is a set of k such
that k + 1 ∉ Lb (i) and k + 1 ∉ Lb (i + 1).
10. Numerical Illustrations for The Periodic Table of Primes
Let λ(i, k) be the integer value of the entry (i, k) in PTP+ (0, 210) if Figure 3, which
is computed based on CTC (1) and the Formula of Primes. For instance, λ(i, k) on
the diagonal dotted points (1, 11), (2, 13), (3,17), (4, 19) and (5, 23) are computed
below.
𝑙(1, 1) = 12, 𝑙(2, 2) = 14, ⋯, 𝑙(23, 23) = 104, ⋯, 𝑙(46, 46) = 200 are converted to
λ(1, 11), λ(2, 13), … λ(23, 17), ⋯, λ(46, 103) , respectively, shown by red numbers as
λ(1, 11) = r1 + 210(𝑙(1, 1) − 1) = 11 + 210(12 − 1) = 2321 = 11 × 211,
λ(2, 13) = r2 + 210(𝑙(2, 2) − 1) = 13 + 210(14 − 1) = 2743 = 13 × 211,
⋮
Similarly,
λ(23, 103) = r23 + 210(𝑙(23, 23) − 1) = 103 + 210(104 − 1) = 21733 = 103 × 211.
λ(46, 199) = r46 + 210(𝑙(46, 46) − 1) = 199 + 210(200 − 1) = 41989 = 199 × 211.
Also, 𝑙(46, 1) = 11, 𝑙(45, 2) = 13, ⋯, 𝑙(1, 46) = 199 are converted to λ(i, k),
respectively, as
λ(46, 10) = r1 + 210(𝑙(46, 1) − 1) = 199 + 210(11 − 1) = 2299 = 11 × 209
λ(45, 12) = r2 + 210(𝑙(45, 2) − 1) = 197 + 210(13 − 1) = 2717 = 13 × 209
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⋮
λ(1, 210) = r46 + 210(𝑙(1, 46) − 1) = 11 + 210(199 − 1) = 41591 = 199 × 209.
Now return to columns 22 and 25 in Figure 2. We note that (i, 23) and (i,24), for i =
1, 2, …, 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
Table 1, 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(i, j) in the CTC and λ(i, k) in the PTP+
demonstrate that (i) many composties such as the digonal and intermediate dotted
points which allocate cyclically at the CTC are also allocatted periodically at the
PTP+. (ii) the PTP is obtained from removing all composites periodically allocated
at PTP+. Therefore, we claim that prime numbers are also generated periodically
since it comes from removing all periodically distributed composites.
Table 1 contains the first 11 columns for the PTP+(0, 210), which is composed of
48 × 11 = 528 elements, represented by λ(i, k) and elaborated below.
(i)
Each entry λ(i, k) which is either a prime or a composite.
(ii) If λ(i, k) is a prime, then λ(i, k) = r𝑖 + 210k , and vice versa. If λ(i, k) is a
composite, then there are q j and q (i,j) such that
λ(i, k) = q j × q 𝐡(𝐢,𝐤) = ri + 210 × (𝑙(i, k + 1) − 1).
Take i = 1, r𝑖 = 11 for instance,
λ(1, 1) = 11 + 210 × (𝑙(1, 2) − 1) = q 2 × q 3 = 13 × 17,
λ(1, 4) = 11 + 210 × (𝑙(1, 5) − 1) = q 5 × q 8 = 23 × 37.
Then, we have λ(1, 0) = 11, λ(1, 2) = 11 × 210(3 − 1) = 431,
λ(1, 3) = 11 + 210 × (4 − 1) = 641, ⋯,
λ(1, 9) = 11 + 210 × (10 − 1) = 1,901.
(iii) Primes λ(i, k) and λ′(i, k) = λ(i, k) + 1 are twin-prime pairs, if and only if r𝑖′ =
2 + r𝑖 , where λ(i, k) = r𝑖 + 210 × k, λ′ (i, k) = r𝑖′ + 210 × k.
For instance, (431, 433), (641, 643), (1,061, 1,063), (1,481, 1,483) are twinprime 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, 2,311]
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
kth column can also be found from Table 1. For instance, the maximum prime
gap for column 2 is 10, occurring at 409 and 419 of r46 and r47 , respectively.
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(v) The PTP(k0, k’) is useful in predicting primes. Examples are given by Nine
Examples for the Predictions of Primes by the PTP.
In summary, the Formula of Primes is checked below:
From Table S1(a), we know that
Lb (1) = {2, 5, 7, 9, 12, ⋯ }
Lb (2) = {7, 10, 12, 13, 14, ⋯ }
Lb (3) = {3, 6, 11, 12, 13, 14, ⋯ }
⋮
From Table 1, we know that
α = r1 + 210k is prime for k = 0, 2, 3, 5, 7, 9, 10
α = r2 + 210k is prime for k = 0, 1, 2, 3, 4, 5, 7, 8, 10
α = r3 + 210k is prime for k = 0, 1, 3, 4, 6, 7, 8, 9
⋮
In general, α = r𝑖 + 210k is prime, if k + 1 ∉ Lb (i)
11. Nine Examples for the Predictions of Primes by the PTP
Example 1: Prediction of the next prime after 1,951
Since 1951 = 61 + 210(10 − 1), from Table 1 there are four
consecutive composites, i.e. 19 × 103, 37 × 53, 13 × 151, and 11 ×
179, right after 1,951. We know the prime next to 1,951 is 1,973.
Example 2: Prediction of the 300th prime
From Expression (3), given b = 1,891, the number of primes within
2, 1,891 is counted as
48
π(1,891) = 4 + 48 × (8 + 1) − ∑|Lb (i)| = 4 + 432 − 152 = 284.
𝑖=1
The 284th prime is 1889, which is located at (i, k) = (47, 8) as shown
in Table 1. Therefore, the 300th prime is counted to be 1,993, at the
location of (i, k) = (23, 9).
Example 3: Prediction of the next twin prime after the twin-prime pair (1277,
1279)
The location of 1,277 is at (i, k) = (3, 6). The twin prime λ(i, k) are
and λ′ = 𝜆(i, k) + 1 expressed as λ(i, k) = r𝑖 + 210(7 − 1). λ′ =
r𝑖′ + 210(7 − 1) and r𝑖′ = r𝑖 + 2.
Therefore, r𝑖 = 29, r𝑖′ = 31, and λ(i, k) = 1,289, λ′ = 1,291.
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Example 4: Predicting the longest same-difference primes
We can predict the longest same-difference prime series in
PTP+(0, 10) from the CTC(basic). Checking row 1 of Table S1(a),
L2311 (1, j) = {𝑙(1, 1), 𝑙(1, 2), 𝑙(1, 3), ⋯ , 𝑙(1, 10), 𝑙(1, 11)}
= {2, 5, 7, 9, 12, 16, 24, 29}
By the mirror effect made in Statement 2 and via L2311 (1, j),
L2311 (46, j) = {11, 12, 14, 15, 16, 19, 21, 24, 25, 33, 35, 15, 24} implies
that 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 do not belong to L2,311 (46, j). As shown on
row 46 of Table S1(b).
Therefore, α is prime, for α = 199 + 210k, k = {0, 1, 2, 3, 4, ⋯ , 9}.
We then find the longest same-difference prime series in
PTP+ [0, 10], or a 10-tuple primes series, to be
(199, 409, 619, 829, 1,039, 1,249, 1,459, 1,669, 1,879, 2,089), as shown
on row 46 of Table 1.
Similarly, we can find the next longest same-difference prime series in
PTP+ (0, 10). That happens at row 9, listed to be (881, 1,091, 1,301,
1,511, 1,721, 1,931, 2,141), as shown on row 9 of Table 1.
Example 5: Visualization of the fundamental theorem of arithmetic
How to factorize 46189 on CTC(2) and to visualize 46,189 on PTP+?
Since 46,189=199+210 × 219, we find that the root of 46,189 is
r46 = 199.
Let 46,189 = q j × (q 𝐡(𝐢,𝐣) + 210(θ − 1)). We need to find
𝑙(46, j) = 1 +
qj ×q𝐡(𝐢,𝐣) −199
210
.
We find that a feasible solution to satisfy the above equation occurs
when j = 32 and h(i, j) = 26. Namely, 𝑙(46, 32) = 𝑙(46, 26) = 77,
with q 32 = 143 and q 26 = 113.
Since 46,189 = q j × (q 𝐡(𝐢,𝐣) + 210(θ − 1)) with θ = 2, we have
46189 = 143 × (113 + 210) = 143 × 323 = 11 × 13 × 323.
On CTC (2), we find q 32 and q (26+48) = q 74 to fit 46,189 = q 32 × q 74,
as shown in Figure S1(a).
On PTP+ (0, 219), we have (i, k) = (1, 74) and
(i′ , k ′ ) = (1, 32), where λ(1, 74) = 323 and
Electronic copy available at: https://ssrn.com/abstract=4742238
24
λ(1, 32) = 143 such that 46189 = λ(1, 74) × λ(1, 32).
Visualization of CTC(2) and PTP+(0, 219) can be seen in Figure S1(b).
Example 6: The maximum prime gap in the interval [10,000, 300,000]
The prime gap in the interval of two primes i, j  is j - i, where
(i)
(ii)
(iii)
i = ri + 210k is a prime rooted to ri ∈ S, and k+1 ∉ Lb (i),
j = ri+n + 210k is a primed rooted to ri+n ∈ S for some n
and k + 1 ∉ Lb (i + n), and
i+m = ri+m + 210k is a composite for m = 1, 2, …, n-1 and
k +1 ∈ Lb (i + m).
In this way, checking L𝑏 (i) for b = 299,881 with b – 211 being a
multiplier of 210, we find that i = 7 and k = 1,340 such that
(i)
(ii)
(iii)
7 = r7 + 210 ×1,340 = 281,431 is a prime, where 1,340+1
= 1,341 ∉ Lb (7),
25 = r25 + 210 ×1,340 = 281,509 is a prime, where 1,340
+1 = 1,341∉ Lb (25), and
7+m = r7+m + 210×1,340 are composites for m = 1, 2, …,
17, where 1,340 +1 ∈ Lb (7 + m).
Therefore, the prime gap within 281,431, 281,509 is 281,509 –
281,431 = 78, which is the maximum prime gap within 10,000,
300,000, where 281,431 is located at the entry (25, 1,340) at
PTP+(47, 1,427) on Figure S2.
Example 7: The largest twin-prime pair in the interval [10,000, 300,000]
We can count the largest twin-prime gap using Expression (4).
Checking PTP+ (47, 1,427), we find it to be (299,699, 299,701), where
299,699 = 29 + 210 × 1,427 and 299,699 = 31 + 210 × 1,427, which
are located at (i, k) = (6, 1,427) and (7, 1,427), respectively. See
Figure S2.
Example 8: How many primes are within the interval [2, 2,311]?
From the CTCs, we count to find ∑48
𝑖=1|Lb (i)| = 188 for b =2,311. From
Expression (3), π(2311) = 4 + 48(10 − 1) − 188 = 248.
Electronic copy available at: https://ssrn.com/abstract=4742238
25
Example 9: How many twin-prime pairs are within the interval [11, 2311]?
Referring to Expression (4), let b = 2311, from CTC, we compute
T(i, i + 1), for i ∈ {1, 3, 6, 9, 13, 16, 22, 24, 30, 33, 37 40, 43, 45, 47} to
reach
T(1,2) = {k | k + 1 ∉ Lb (1), k + 1 ∉ Lb (2)} = {0, 2, 3, 5, 7,10}
Similarly, we have
T(3, 4) = {0, 1, 4, 6, 7, 8}, I(6, 7) = {0, 1, 3, 6}, and
T(9, 10) = {0, 2, 4, 5, 6, 7, 8, 9}, and T(47, 48) = {0, 1, 4, 10}.
Therefore, the number of twin-prime pairs is
π∗ (2311) = 3 + ∑i∈w|T(i, i + 1)| = |T(1, 2)| + |T(3, 4)| + |T(6, 7)|
+ ⋯ + |T(47, 48)| = 3 + 6 + 6 + ⋯ + 4 = 69.
12. Justification of Claim 4 in Discussion
Given a set of Lb (i, j + 48(θ − 1)), if there exist 𝑙(i, j∗ ) and 𝑙(i, j∗ + 1)with 11 ≤ j∗ ≤
48θ such that 𝑙 (i, j∗ ), 𝑙 (i, j∗ + 1) ∈ Lb (i, j + 48(θ − 1)), for all 1 ≤ i ≤ 48, then the
maximum prime gap within an interval [11, b] is 420. This implies that there exists
no prime between α = r1 + 210 (𝑙(1, j∗ ) − 1) and α = r48 + 210 (𝑙(48, j∗ + 1) − 1).
If there exist 𝑙(i, j∗ ), 𝑙(i, j∗ + 1), 𝑙(i, j∗ + 2) with 11 ≤ j∗ ≤ 48θ such that
𝑙(i, j∗ ), 𝑙(i, j∗ + 1), 𝑙(i, j∗ + 2) ∈ Lb (i, j + 48(θ − 1)) for all 1 ≤ i ≤ 48, then the
maximum prime gap within an interval [11, b] is 630. This implies that there exists
no prime between α = r1 + 210(𝑙(1, j∗ ) − 1) and α′ = r48 + 210 (𝑙(48, j∗ + 2) − 1).
Electronic copy available at: https://ssrn.com/abstract=4742238
26
q𝑗
r𝑖
𝑙(𝑖, 𝑗)
r46
q26
q32
q74
=113
=143
=323
𝑙(46, 26)
𝑙(46, 32)
per
r𝑖
k
𝜆(𝑖,k)
r1
r2
r3
r4
0
26
32
143
74
r46
46189
(b)
Figure S1: Visualization of 46189 and its factors
root i
r𝑖
period k
1
2
11
13
⋮
⋮
6
7
29
31
⋮
109
25
⋮
𝜆(𝑖,k)
max prime gap
47
⋯
1340
⋯
1427
299699
281431
219
323
⋮
𝑙(46, 74)
(a)
iod
299701
281509
78
Figure S2: Maximum prime gap and the largest
twin-prime pair in PTP[10000, 300000]
Electronic copy available at: https://ssrn.com/abstract=4742238
27
Table S1(a): A basic composite table CTC(basic)
j
ent q
segm
root i
1
r𝑖
𝑙(𝑖, 𝑗)
11
𝑗
1
2
3
4
5
6
7
8
9
10
11
12
11
13
17
19
23
29
31
37
41
43
47
53
⋯
23
24
⋯
46
47
48
⋯ 103 107 ⋯ 199 209 211
12
2
2
9
5
16
7
5
7
29
24
33
2
13
10
14
13
7
12
24
25
36
23
38
41
34
24
84
199
12
75
31
197
14
3
17
6
12
18
3
3
11
30
24
14
13
28
4
19
4
11
12
20
10
19
17
18
30
22
45
36
74
32
193
18
37
22
86
191
20
5
23
11
9
17
16
24
6
22
6
21
40
32
39
21
87
187
24
6
29
5
6
16
10
22
30
14
25
28
24
36
42
71
35
181
30
7
31
3
5
10
8
6
9
32
19
3
33
6
43
19
89
179
32
8
37
8
2
9
2
4
4
24
38
10
17
10
46
69
37
173
38
9
41
4
13
14
17
18
20
29
26
42
35
44
48
68
38
169
42
10
43
2
12
8
15
2
28
16
20
17
44
14
49
16
92
167
44
11
47
9
10
13
11
16
15
21
8
8
19
48
52
15
93
163
48
12
53
3
7
12
5
14
10
13
27
15
3
5
54
65
41
157
54
13
59
8
4
11
18
12
5
5
9
22
30
9
4
12
96
151
60
14
61
6
3
10
16
19
13
23
3
38
39
26
5
63
43
149
62
15
67
11
13
4
10
17
8
15
22
4
23
30
8
10
98
143
68
16
71
7
11
9
6
8
24
20
10
36
41
17
10
9
99
139
72
17
73
5
10
3
4
15
3
7
4
11
7
34
11
60
46
137
74
18
79
10
7
2
17
13
27
30
23
18
34
38
14
7
101
131
80
19
83
6
5
7
13
4
14
4
11
9
9
25
16
6
102
127
84
20
89
11
2
6
7
2
9
27
30
16
36
29
19
56
50
121
90
21
97
3
11
16
18
7
12
6
6
39
29
3
23
54
52
113
98
22
101
10
9
4
14
21
28
11
31
30
4
37
25
53
53
109 102
23
103
8
8
15
12
5
7
29
25
5
13
7
26
104 107
107 104
47
209
1
7
3
1
8
25
22
3
33
17
15
26
26
210 210
Note: Numbers in red — for diagonal effect. Numbers in green — for intermediate effect.
Electronic copy available at: https://ssrn.com/abstract=4742238
80
28
Table S1(b): A complementary composite table CTC (complementary)
j
ent q
segm
root i
24
r𝑖
𝑙(𝑖, 𝑗)
107
𝑗
1
2
3
4
5
6
7
8
9
10
11
12
11
13
17
19
23
29
31
37
41
43
47
53
⋯
23
24
⋯
46
47
48
⋯ 103 107 ⋯ 199 209 211
4
6
3
8
19
23
3
13
37
31
41
28
25
109
2
5
14
6
3
2
21
7
12
40
11
29
103 108
103 108
26
113
9
3
2
2
17
18
26
32
3
15
45
27
121
1
12
12
13
22
21
5
8
26
8
19
28
127
6
9
11
7
20
16
28
27
33
35
23
38
98
6
83
128
29
131
2
7
16
3
11
3
2
15
24
10
10
40
97
7
79
132
30
137
7
4
15
16
9
27
25
34
31
37
14
43
44
62
73
138
31
139
5
3
9
14
16
6
12
28
6
3
31
44
95
9
71
140
32
143
1
1
14
10
7
22
17
16
38
21
18
46
94
10
67
144
33
149
6
11
13
4
5
17
9
35
4
5
22
49
41
65
61
150
34
151
4
10
7
2
12
25
27
29
20
14
39
50
92
12
59
152
35
157
9
7
6
15
10
20
19
11
27
41
43
53
39
67
53
158
36
163
3
4
5
9
8
15
11
30
34
25
47
3
89
15
47
164
37
167
10
2
10
5
22
2
16
18
25
43
34
5
88
16
43
168
38
169
8
1
4
3
6
10
3
12
41
9
4
6
36
70
41
170
39
173
4
12
9
18
20
26
8
37
32
27
38
8
35
71
37
174
40
179
9
9
8
12
18
21
31
19
39
11
42
11
85
19
31
180
41
181
7
8
2
10
2
29
18
13
14
20
12
12
33
73
29
182
42
187
1
5
1
4
23
24
10
32
21
4
16
15
83
21
23
188
43
191
8
3
6
19
14
11
15
20
12
22
3
17
82
22
19
192
44
193
6
2
17
17
21
11
2
14
28
31
20
18
30
76
17
194
45
197
2
13
10
13
12
6
7
2
19
6
7
20
29
77
13
198
46
199
11
12
16
11
19
14
25
33
35
15
24
21
80
24
11
200
48
211
10
6
14
18
15
4
9
34
8
26
32
27
77
27
208 212
51
55
101 110
31
50
56
97
114
35
48
58
89
122
Note: Numbers in red — for diagonal effect. Numbers in green — for intermediate effect.
Electronic copy available at: https://ssrn.com/abstract=4742238
29
Table S2(a): Upper rows of CTC(2)
j
ent q
segm
root i
1
r𝑖
𝑙(𝑖, 𝑗)
11
𝑗
49
50
221
223
⋯
Table S2(b): Lower rows of CTC(2)
96
j
ent q
segm
root i
24
r𝑖
𝑙(𝑖, 𝑗)
107
𝑗
49
50
221
223
71
95
78
223
19
225
25
109
31
2
13
183
3
17
103
191
26
113
172
44
4
19
63
174
27
121
12
199
5
23
204
140
28
127
113
48
6
29
84
89
29
131
33
114
7
31
44
72
30
137
134
63
8
37
145
21
31
139
94
46
9
41
65
210
32
143
14
12
149
115
184
10
43
25
193
33
11
47
166
159
34
151
75
167
157
176
116
12
53
46
108
35
13
59
147
57
36
163
56
65
14
61
107
40
37
167
197
31
15
67
208
212
38
169
157
14
16
71
128
178
39
173
77
203
17
73
88
161
40
179
178
152
18
79
189
110
41
181
138
135
19
83
109
76
42
187
18
84
20
89
210
25
43
191
159
50
193
119
33
21
97
50
180
44
22
101
191
146
45
197
39
222
129
46
199
220
205
47
209
20
120
48
211
202
104
23
103
151
Electronic copy available at: https://ssrn.com/abstract=4742238
⋯
96
30
Table S3: List of L44521((1, j + 48(θ-1)), for θ = 1, 2, …, 19
θ=1
j
qj
qh(𝑖,𝑗)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101
103
107
109
113
121
127
131
137
139
143
149
151
157
163
167
169
173
179
181
187
191
193
197
199
211
17
13
89
37
109
41
23
31
137
103
127
139
131
113
151
167
149
157
19
143
181
47
163
29
67
191
53
61
43
59
37
79
71
83
107
73
179
187
169
101
173
121
197
193
209
47
48
209
211
199
11
∆𝑖,θ
𝑙(1,j)
12
2
2
9
5
16
7
5
7
29
24
33
40
39
37
52
59
57
63
9
67
88
24
84
16
37
111
33
39
29
40
67
57
52
63
84
59
145
155
145
88
155
111
182
182
199
θ=2
θ=3
θ = 19
𝑙(1, j+48)
= 𝑙(1, j)+qh(𝑖,𝑗)
𝑙(1, j+96)
= 𝑙(1, j)+2qh(𝑖,𝑗)
𝑙(1, j+864)
= 𝑙(1, j)+18qh(𝑖,𝑗)
223
19
15
98
42
125
48
28
38
166
127
160
179
170
150
203
226
206
220
28
210
269
71
247
45
104
302
86
100
72
99
104
136
123
146
191
132
324
342
314
189
328
232
379
375
408
434
36
28
187
79
234
89
51
69
303
230
287
318
301
263
354
393
355
377
47
353
450
118
410
74
171
493
139
161
115
158
141
215
194
229
298
205
503
529
483
290
501
353
576
568
617
⋯
3810
308
236
1611
671
1978
745
419
565
2495
1878
2319
2542
2397
2071
2770
3065
2739
2889
351
2641
3346
870
3018
502
1243
3549
987
1137
803
1102
733
1479
1330
1557
2010
1373
3367
3521
3187
1906
3269
2289
3728
3656
3961
199
12
398
23
597
34
3781
210
2
15
28
210
∆𝑖,θ=min{𝑙(i, j+48(θ-1)}, i=1
j
Electronic copy available at: https://ssrn.com/abstract=4742238
31
Table S4: The Periodic Table of Primes of various intervals
97
root i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
r𝑖
period k
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101
103
107
109
113
0
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101
103
107
109
113
11x11
11x11
127
131
137
139
127
131
137
139
11x13
11x13
149
151
157
163
167
149
151
157
163
167
13x13
13x13
173
179
181
173
179
181
11x17
11x17
191
193
197
199
191
193
197
199
11x19
11x19
211
211
99
100
101
102
103
104
105
106
⋯
21011 21221
21851
21013
21433
22063
49
51
53
54
55
56
57
58 ⋯
96
20807 21017 21227
21647
22067
20389 20599 20809 21019
21649 21859
10091 10301
11981
19961
20183 20393
21023 11351
21863 22073
10093 10303 10513
11353
19963
20399 10723
22079
2
3
4
5 20611
6 10937
7
8210319 10 ⋯
48
11777
11987
21661 21871
10099
11779
20407 10729 10939 11149 21247
10103431
10313
11783 21881 22091
19973
20201
641 20411 10733
1061
1481
1901
11159
11789
19979
21673
22093
223 433 64310529
853 10739
1063 10949
1483
169311369 11579 9883
10111
10321
10531
11161
227
647 857 206271277 1487 1697 1907 214679887
10957
11587 21683 12007
229 439
859 1069 1279
1489 1699
21893
10331
11171
11801 12011 22109
19991
233 443
653 863 206391283
1493
1913
20219
20849
21059 21269
10333
11173
19993
239 449
659 20431 10753
170911383 11593
206411289 1499
21061
21481
22111
10337
11177
11597
11807
19997
241
661
1291
20857 21067 21277 214879901
10133457
10343
10973
11393 214919907
11813
877 1087 1297
20231 20441
21701 21911
10139
10979
251 46120233 10559
881 1091 1301
1511 172111399
1931 21493
20443
21283
22123
10141463 673 883 10771
20011
1093 1303 1523 1723 1933 21499 11821
22129
10357
10567
10987
11197
11617
11827
12037
257 467 677 887 1097
206631307
20873
21503 21713
22133
10151
10781
11831 12041
20021
263
683
1103 20879
152321089
173311411 11621 9923
20249
21929
10993
11833
20023
20887
21727 12043
21937 22147
269 479
1109 1319
1949 215179929
10159
10369
10789
11839 12049
20029
20261
206811321 153121101
271
691
1741 1951 215219931
10163
11003
11213
11423
11633
21943 22153
277 487
907 1117 1327
1747 21313 21523
10169491 70110589
20477
281
911 10799 20897 21107 21317
9941
21737
22157
10177 20269 10597
20047
283
1543 175311437
20479 1123 20899
21319 21529 21739
22159
10181499
10391
12071
20051
70910601
919 1129
1549 1759 21323
9949
20483
20693 20903
11443
11863
12073
293 503
155321121 1973
21751 21961 22171
11447
50920287
71910607
929 20707 11027
1559
1979 11657 11867
21757
10399
11239
307
727 937
1567
1777 21341
1987
9967
20921
10193
11243 21347 21557 21767 21977
20063
311 52120297 10613
941 1151
20507
207171361 1571
10831
11251
20071
313 523 733 20509 1153
1783 1993 215599973
20719 20929 21139
22189
10627
11047 11257
317
947 10837 1367
178711467
1997 11677
21143
21563 11887
21773 12097 22193
10211
11261
12101
73910631
1579
178911471
1999 11681
20939
21149
21569
10427
11057
11897 12107
743 20521
953 10847
1163
1583
2003
207311373
21991
10429
11059
20089
20947 21157
21577 21787 12109
21997
331 541
75110639 1171 1381
1801 2011 11689
10223547
10433
11273
11903 12113
20323
20743
21163 11483
22003
337
757 20533
967 10853
1597
2017
11279
20327
20747 11069
21377 11699
21587 11909 12119
761 971 10859
1181
1601
181111489
10651
20101
20749 11071
20959
21379
21589
21799
347 557
977 10861
1187
160721169 11491
2027 11701
10007
10657
10867
11287
11497
20107
20333
20963
21383
21803 22013
349
769 20543 207531399
1609
2029
10009
10243563
10453
20113
20549
20759 11083
22229
353
77310663
983 1193
161321179
182311503 21599 11923
10247
10457
11087
20117
20341 10667
20551
21391
21601 11927
359 569
1409
1619
2039 11717
10459
11299
20347 991 1201
21187
571
1621
1831 21397 11719 21817 22027
10253
10463
20123
20771 11093
20981
21821 12143
22031
367 577
787 997 10883
162721191 21401 21611 11933
10259
20129
20353 20563 10889
207731423
20983 21193 11519
373
1213
2053 21613 11939 12149
10891
11311
11941 22037 22247
21617
58720357
797
1217 1427 1637
1847 21407 11731
10037
10267
10477
11317 11527
12157
20359 10687
22039
379
1009
1429
10039
10271
10691
11321
207891433
21419
21839 12161 22259
383 59320369 1013 1223
2063
10273599 809 1019 10903
11113
20143
21001 21211 2069 11743 11953
21841 12163
22051
389
1229 1439
10487
20147
601
811 1021 1231 11117
1861
10909
11119
11329
11959
20149
397 607
1237 1447
1657
1867
10289
10499
10709
11549
11969
401
821 1031
1451
1871 2081
10061
10501
11131
11971
20161
613
82310711
1033
1453
1663 187311551
2083
617 827
1667 1877 2087
10067
409 619 829 1039 1249 1459 1669 1879 2089
10069
419
839 1049 1259
1889 2099
10079 PTP(0, 48)
421 631
1051
1471
20173
50
20177
𝜆(𝑖,k)
98
20593
52
Electronic copy available at: https://ssrn.com/abstract=4742238
144
30047
30059
30071
30089
30091
30097
30103
30109
30113
30119
30133
30137
30139
⋯
30161
30169
30181
30187
30197
30203
30211
30223
PTP(97, 144)
30241
PTP(49, 96)