This study proposes a novel way, called Periodic-Table-of-Primes (PTP) method, to list all primes within a pre-specified integer interval. The deduction of this method is based on the following observations:

By choosing the first four primes, 2, 3, 5 and 7, we can group every $2\times 3\times 5\times 7=210$ numbers in one period starting from 11, i.e., ${\left[11,220\right]}_N,{\left[221,430\right]}_N,{\left[431,640\right]}_N,\cdots$. Within each of these periods, there are 48 numbers without factors of 2, 3, 5 and 7. In particular, in ${\left[11,220\right]}_N$, there exist 48 numbers called roots, and we denote

$\begin{array}{l}S=\{r_1=11,r_2=13,r_3=17,r_4=19,r_5=23,r_6=29,r_7=31,r_8=37,r_9=41,r_{10}=43,\\ r_{11}=47,r_{12}=53,r_{13}=59,r_{14}=61,r_{15}=67,r_{16}=71,r_{17}=73,r_{18}=79,r_{19}=83,\\ r_{20}=89,r_{21}=97,r_{22}=101,r_{23}=103,r_{24}=107,r_{25}=109,r_{26}=113,r_{27}=121,\\ r_{28}=127,r_{29}=131,r_{30}=137,r_{31}=139,r_{32}=143,r_{33}=149,r_{34}=151,r_{35}=157,\\ r_{36}=163,r_{37}=167,r_{38}=169,r_{39}=173,r_{40}=179,r_{41}=181,r_{42}=187,r_{43}=191,\\ r_{44}=193,r_{45}=197,r_{46}=199,r_{47}=209,r_{48}=211\}.\end{array}$

Theorem 1. For any number $\alpha \in \text{Λ}$, there exists a unique $r_i\in S$ and $k\in N_0$ such that

$\begin{array}{c}\alpha =r_i+210k\end{array}$. (1)

From Theorem 1, we know that each number $\alpha \in \text{Λ}$ has a root and a period. In other words, $\alpha$ is the $k^{\text{th}}$ descendant of $r_i$.

From notations indicating that $Q=S$ and $q_j=r_j$, $j=1,2,\cdots ,48$, we can also know that any factor of $\alpha$can be expressed as $q_j+210\theta$, for some $q_j\in Q$ and $\theta \in N_0$. The following observation further analyzes the structure of $\alpha$’s factors.

Given $q_j\in Q$ and $r_i\in S$, denote $q_{\hat{j}|i}$ as the unique element in $Q$ such that $q_j\times q_{\hat{j}|i}-r_i$ is a multiplier of 210, where $i,j\in \left\{1,2,\cdots ,48\right\}$. Here $\left(q_j,q_{\hat{j}|i}\right)$ is called a factor-pair with respect to $r_i$. We also observe there exists an integer

$\begin{array}{c}{t}_{j|i}=\frac{q_j{q}_{\hat{j}|i}-r_i}{210}\end{array}$ (2)

to specify the generation of $r_i$ in which the descendant of $q_j$ and $q_{\hat{j}|i}$, i.e., $q_j\times q_{\hat{j}|i}$ lies.

Collecting all $\left(q_j,q_{\hat{j}|i}\right)$ and $t_{j|i}$ for all $i,j\in \left\{1,2,\cdots ,48\right\}$ together, we then form a Factor-Pair Table with generation information. It is a $48\times 48$ table made of element $\left(q_j,q_{\hat{j}|i}\right),t_{j|i}$.

We denote each column of the table as $V_j$, where

$V_j={\left(\left(q_j,q_{\hat{j}|1}\right),t_{j|1};\cdots ;\left(q_j,q_{\hat{j}|i}\right),t_{j|i};\cdots ;\left(q_j,q_{\hat{j}|48}\right),t_{j|48}\right)}^{\text{T}}$ for $j=1,2,3,\cdots ,48$.

Take $j=1$ for instance,

$V_1={\left(\left(11,211\right),11;\left(11,173\right),9;\left(11,97\right),5;\left(11,59\right),3;\cdots ;\left(11,191\right),9\right)}^{\text{T}}$,

where $\left(q_1,q_{\hat{1}|1},t_{1|1}\right)=\left(q_1,q_{48},t_{1|1}\right)=\left(11,211,11\right)$, as shown in Table 1 .

The Factor-Pair Table with generation information has the following properties:

1) In each row of the table, there are 48 factor-pairs.

2) If $q_j=q_{\hat{j}|i}$, then $\left(q_j,q_{\hat{j}|i}\right)$ is called a co-factor-pair. Co-factor-pairs only exist in 6 rows of the table. In each row corresponding to $r_{18}=79$, $r_{25}=109$, $r_{27}=121$, $r_{34}=151$, $r_{38}=169$ and $r_{48}=211$, there are 8 co-factor-pairs among the 48 factor-pairs.

3) If $\left(q_j,q_{\hat{j}|i}\right)$ is not a co-factor-pair, then $\left(q_{\hat{j}|i},q_j\right)$ is a factor-pair with the same generation $t_{j|i}$ as $\left(q_j,q_{\hat{j}|i}\right)$.

4) For any $j$ and $i$, we have $t_{j|i}\le r_j$.

We found that the factors of a composite number $\alpha$ are closely related to factor-pairs in the Factor-Pair Table with generation information:

$\frac{\alpha }{q_j+210\theta }$ is a descendant of $q_{\hat{j}|i}$, i.e.,

$\begin{array}{c}\frac{\alpha }{q_j+210\theta }=q_{\hat{j}|i}+210\hat{\theta }\end{array}$. (3)

Theorems 1 and 2 confirm that any composite number $\alpha$ without factors of 2, 3, 5 and 7 can be written as

$\begin{array}{c}\alpha =r_i+210k=\left(q_j+210\theta \right)\left(q_{\hat{j}|i}+210\hat{\theta }\right)\end{array}$, (4)

where $\theta ,\hat{\theta }\in N_0$ and $j\in \left\{1,2,\cdots ,48\right\}$.

From Equation (4), we have

$\begin{array}{c}{r}_i+210k=q_j{q}_{\hat{j}|i}+210\theta q_{\hat{j}|i}+210\hat{\theta }{q}_j+{210}^2\theta \hat{\theta }\end{array}$. (5)

Combining Equations (2) and (5), we get

$\begin{array}{c}{t}_{j|i}=k-\theta q_{\hat{j}|i}-\hat{\theta }{q}_j-210\theta \hat{\theta }\end{array}$. (6)

When $j$ is fixed, Equation (6) is a quadratic equation with only two unknowns $\theta$ and $\hat{\theta }$ in $N_0$.

We then propose an algorithm to solve the equation.

Without loss of generality, we assume $q_j+210\theta \le q_{\hat{j}|i}+210\hat{\theta }$, since we can replace $q_j$ and $\theta$ with $q_{\hat{j}|i}$ and $\hat{\theta }$ correspondingly if $q_j+210\theta >q_{\hat{j}|i}+210\hat{\theta }$. Given $\alpha =\left(q_j+210\theta \right)\left(q_{\hat{j}|i}+210\hat{\theta }\right)$, we see

$\begin{array}{c}{q}_j+210\theta \le \sqrt{\alpha }\end{array}$. (7)

Therefore,

$\begin{array}{c}\theta \le \frac{\sqrt{\alpha }-q_j}{210}\le \frac{\sqrt{\alpha }}{210}\end{array}$. (8)

We can derive a direct corollary from Theorem 3:

Corollary 1. Checking whether a number $\alpha =r_i+210k\in \text{Λ}$ is a composite number is equivalent to checking if

$\begin{array}{c}\hat{\theta }=\frac{k-t_{j|i}-\theta q_{\hat{j}|i}}{q_j+210\theta }\end{array}$ (9)

is an integer for some $\theta \in \left\{0,1,\cdots ,\lceil \frac{\sqrt{\alpha }}{210}\rceil \right\}$.

Based on this Corollary, we can develop a new algorithm for primality test:

The following theorem gives the complexity of Algorithm 1:

However, to list all primes within a pre-specified interval, the above algorithm for primality test costs too much time. Therefore, we need to find a more efficient way to locate primes.

According to the 4th property of the Factor-Pair Table with generation information, we know

$\begin{array}{c}{t}_{j|i}+r_j\le 2r_j\end{array}$. (10)

To represent the generation information of each factor pair more clearly, we convert each column of the Factor-Pair Table with generation information, $V_j$, into a basic binary matrix of composites $B_j^o={\left[b_j^0\left(i,k\right)\right]}_{48\times 2r_j}$ for $i=1,2,\cdots ,48$, $k=0,1,2,\cdots ,2r_j-1$, where $b_j^0\left(i,k\right)\in \left\{0,1\right\}$. In this matrix, $b_j^0\left(i,k\right)=1$ indicates that the $k^{\text{th}}$ descendant of $r_i$ is a composite with a factor being $q_j$.

Specifically, we specify $b_j^0\left(i,k\right)$ following the rule below:

The entry $b_j^0\left(i,k\right)$ is set as 1 only under two cases:

Case 1 ( $i=j$): if $k=r_i$ or $k=0$, $b_j^0\left(i,k\right)=1$.

Case 2 ( $i\ne j$): if $k=t_{j|i}$ or $k=t_{j|i}+r_j$, $b_j^0\left(i,k\right)=1$.

Otherwise, $b_j^0\left(i,k\right)=0$.

Inequality (10) guarantees that the rule is properly defined.

We observe the following properties in $B_j^0$:

1) There are exactly two non-zero entries for each $i$, which means either Case 1 or Case 2 always sets two entries in $B_j^0$ as 1.

2) Every matrix $B_j^0$ consists of two identical submatrices of the size $48\times r_j$ placed side by side.

3) Each non-zero entry in the submatrix mentioned in (2) corresponds to a factor-pair $\left(q_j,q_{\hat{j}|i}\right)$, $i=1,2,\cdots ,48$ with its generation $t_{j|i}$.

Figure 1 shows how to convert $V_1$ to $B_1^0$. For instance,

we set $b_1^0\left(1,0\right)=b_1^0\left(1,11\right)=1$, moreover,

since $t_{1|2}=9,b_1^0\left(2,9\right)=b_1^0\left(2,20\right)=1$;