Instrumentals behind embryo and cancer: a platform for prospective future in cancer research

Kishore Kumar Meenakshi Sundaram; Giridharan Bupesh; Konda Mani Saravanan; Kishore Kumar Meenakshi Sundaram; Giridharan Bupesh; Konda Mani Saravanan

doi:10.3934/molsci.2022002

AIMS Molecular Science

2022, Volume 9, Issue 1: 25-45. doi: 10.3934/molsci.2022002

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Review

Instrumentals behind embryo and cancer: a platform for prospective future in cancer research

1.
Research and Publication Wing, Bharath Institute of Higher Education and Research, Chennai – 600073, Tamil Nadu, India
2.
Department of Forest Science, Nagaland University, Hqrs. Lumami, Zunheboto, Nagaland – 798627, India

Received: 21 January 2022 Revised: 16 March 2022 Accepted: 27 March 2022 Published: 31 March 2022

The cancer cells could be celled biomass without normal cellular regulation. They bypass most of the signaling pathways leading to programmed cell division. On the other hand, the embryos are highly regulated, giving rise to the whole organism based on the planned regulation. Understanding the bridge concepts between them might be an interventional art for discovering valuable cancer drugs. The present review highlighted the most similarities between them and recent literary works.

Keywords:

Citation: Kishore Kumar Meenakshi Sundaram, Giridharan Bupesh, Konda Mani Saravanan. Instrumentals behind embryo and cancer: a platform for prospective future in cancer research[J]. AIMS Molecular Science, 2022, 9(1): 25-45. doi: 10.3934/molsci.2022002

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Abstract

1. Introduction

Graph plays a dynamic role in various sciences such as physics, biology, chemistry, and computer science ^[2,3,4,5,8]. It is used in various frameworks related to social and information systems ^[4] and also solves many issues related to everyday life. In physics, there are various circuits constructed by considering different graphs ^[5]. The atomic number of many molecules is evaluated by using group symmetry graphs that are still unknown a few years ago ^[3]. In computer science, many problems have been discussed using graphs that were not easy to visualize earlier. For discrete mathematics and combinatorics, the applications of number theory and graph theory are of crucial importance. In this work, we employ number theory to investigate the special classes of graphs.

Rogers ^[18] discussed the action of a quadratic map on multiplicative groups under modulo a prime $p$ by using the associated directed graph for which there is an edge from each element to its image. He established a formula to decompose a graph into cyclic components with their attached trees. The necessary and sufficient conditions for the existence of isolated fixed points have also been established. Somer and Krizek ^[19] studied the structures of graphs of quadratic congruences for composite modulus. Mahmood and Ahmad ^[10,11] proposed many new results of graphs over residues modulo prime powers. Haris and Khalid ^{[12,13,14,15]} investigated the structure of power digraphs associated with the congruence $x^{n}\equiv y\; (mod\; m)$ . Meemark and Wiroonsri ^[9] discussed the structure of $G(R, k)$ using a quadratic map, where $R$ is the quotient ring of polynomials over finite fields and $k$ is the modulus. Wei and Tang ^[20] introduced the concept of square mapping graphs of the Gaussian ring $Z_{m}[i]$ . Ali et al. ^[21,22] introduced new labeling algorithm on various classes of graphs with applications. Some basic and useful result discussed in ^{[1,6,7,16,23,24]} as well.

Let $p$ be a prime and $a$ an integer coprime to $p$ . Then $a$ is called a quadratic residue $\; (\text{mod}\; p)$ if and only if the congruence $x^{2}\equiv a\; (\text{mod}\; p)$ has a solution. Otherwise, $a$ is called quadratic non-residue $\; (\text{mod}\; p)$ . Two non-zero integers $x$ and $y$ are called zero divisors in the ring $Z_{m}$ if and only if $xy\equiv 0\; (\text{mod}\; m)$ ^[17]. Recall that a graph $\overbrace{G}(2, m)$ , whose vertices are elements of ring $Z_{m}$ and there will be an edge between $\overline{x}$ and $\overline{y}$ $(\overline{x}\neq \overline{y})$ if $\overline{x}^{2}\equiv \overline{y}^{2}\; (\text{mod}\; m)$ then, $\overbrace{G}(2, m)$ is termed as a quadratic graph. For $m = 30$ vertex set is

$Z_{n} = \{\overline{0}, \overline{1}, \overline{2}, \overline{3}, \overline{4}, \overline{5}, \overline{ 6}, \overline{7}, \overline{8}, \overline{9}, \overline{10}, \overline{11}, \overline{12}, \overline{13}, \overline{14}, \overline{15}, \overline{16}, \overline{17}, \overline{18}, \overline{19}, \overline{20}, \overline{ 21}, \overline{22}, \overline{23}, \overline{24}, \overline{25}, \overline{26}, \overline{27},$ $\overline{28}, \overline{29} \}$ , by solving the congruences $\overline{x}^{2}\equiv \overline{y}^{2}\; (\text{mod}\; 30)$ for each $\overline{x},$ $\overline{y}\in Z_{n}$ then, there are 2, 4, 6 copies of $K_{1}$ , $K_{4}$ and $K_{2}$ , respectively as shown in . We note that each copy of $K_{2}$ and $K_{4}$ has equal sum 30 and 60, respectively. Here $K_{n}$ is a complete graph obtained if each node connected with every other node except itself ^[4].

Figure 1.

$\overbrace{G}(2, 30) = 2K_{1}\bigoplus 6K_{2}\bigoplus 4K_{4}$ .

DownLoad: Full-Size Img PowerPoint

Theorem 1.1. ^[17] Let $p$ be an odd prime, $k$ be a positive integer, and $a$ an integer such that $(a, p) = 1$ . Then

1. The congruence $x^{2}\equiv a\; ({mod}\; p^{k})$ has either no solution or exactly two incongruent solutions modulo $p^{k}.$

2. The congruence $x^{2}\equiv a\; ({mod}\; p^{k})$ has no solution if $a$ is quadratic non-residue of $p$ and exactly two incongruent solutions modulo $p$ if $a$ is quadratic residue of $p$ .

Theorem 1.2. ^[17] Let $a$ be an odd number. Then we have the following:

1. The congruence equation $x^{2}\equiv a\; ({mod}\; 2)$ has the unique solution if and only if $x\equiv 1 \; ({mod}\; 2).$

2. The congruence equation $x^{2}\equiv a \; ({mod}\; 4)$ either has no solution if $a\equiv 3\; \; {mod}\; 4)$ or has two solutions $x \equiv 1, 3 \; \; ({mod}\; 4)$ if $a\equiv 1 \; ({mod}\; 4)$ .

3. When $k\geq 3,$ the equation $x^{2}\equiv a \; ({mod}\; 2^{k})$ either has no solution if $a\not\equiv 1\; ({mod}\; 8)$ ; or has four solutions $x_{1}, -x_{1}, x_{1}+2^{k-1}, -(x_{1}+2^{k-1})$ if $a \equiv 1 \; ({mod}\; 8)$ .

2. Quadratic residues graphs over $Z_{2^{\beta}}$ and $Z_{q^{\beta}}$

In this section, we characterize quadratic residue graphs for some well-known classes of integers $2^{\beta}$ and $q^{\beta},$ for each positive integer $\beta$ and odd prime $q$ .

Theorem 2.1. Let $m = 2^{\beta}$ be an integer. Then

$\overbrace{G}(2, 2^{\beta}) = \left\{ \begin{array}{ll} 2K_{\beta}, & {\rm\; if\; }\beta = 1, 2, \\ 2K_{2}\bigoplus K_{4}, & {\rm\; if\; }\beta = 3, \\ 4K_{4}, & {\rm\; if\; }\beta = 4, \\ 6K_{4}\bigoplus K_{8}, & {\rm\; if\; }\beta = 5, \\ 4 K_{8}\bigoplus 8 K_{4}, & {\rm\; if\; }\beta = 6, \\ 6K_{8}\bigoplus K_{16}\bigoplus 16 K_{4}, & {\rm\; if\; }\beta = 7, \\ 8K_{8}\bigoplus 4 K_{16}\bigoplus 32 K_{4}, & {\rm\; if\; }\beta = 8, \\ \bigoplus\limits_{i = 1}^{\frac{\beta -7}{2}}2^{\beta-5-2(i-1)}K_{2^{3+(i-1)}}\\ \bigoplus 6K_{2^{(\beta-1)/2}}\bigoplus K_{2^{(\beta+1)/2}}\bigoplus 2^{\beta-3}K_{4}, & {\rm\; if\; }\beta\geq 9, {\rm\; and\; }\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \beta\equiv 1\; ({mod}\; 2), \\ \bigoplus\limits_{i = 1}^{\frac{\beta -6}{2}}2^{\beta-2i-3}K_{2^{2+i}} \bigoplus 2^{2}K_{2^{\beta/2}}\bigoplus 2^{\beta-3}K_{4}, & {\rm\; if\; }\beta\geq 10, {\rm\; and\; } \\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \beta\equiv 0\; ({mod}\; 2), \\ \end{array} \right.$

Proof. We discuss two cases to prove this theorem, the zero-divisors and unit elements of the ring $Z_{m}$ . Firstly, we discuss zero-divisors, let $S = \{2m | m = 0, 1, 2, \cdots, 2^{\beta-1}~~-1\}$ be the set of all zero-divisors of $Z_{2^{\beta}}$ with including zero for each positive integer $\beta$ . To find the number of solutions of $\eta^{2}\equiv \beta^{2}\; (\text{mod} \; 2^{\beta})$ , we start from $\eta^{2}\equiv \beta^{2}\; (\text{mod} \; 2)$ , in this case just $S = \{0\}$ . Therefore, $\eta^{2}\equiv 0\; (\text{mod} \; 2)$ has only one solution which is zero, but by the definition of quadratic zero-divisors graph there will be a no loop, so $\overbrace{G}(2, 2) = K_{1}$ . For $\beta = 2$ , $\eta^{2}\equiv 0\equiv2^{2}\; (\text{mod} \; 2^{2})$ has two solution namely $\eta = 0, 2$ , then $\overbrace{G}(2, 2^{2}) = K_{2}$ . If $\beta = 3$ , then there are two congruences

$\eta^{2}\equiv 0\; (\text{mod} \; 2^{3}), \; \; \text{and} \; \; \eta^{2}\equiv 4\; (\text{mod} \; 2^{3}).$

The roots of these congruences are $\eta = 0, 4$ , and $\eta = 2, 6,$ respectively. Thus, there exist two copies of $K_{2}$ . For $\beta = 4$ , we have

$\eta^{2}\equiv 0\; (\text{mod} \; 2^{4}), \; \; \text{and} \; \; \eta^{2}\equiv 4\; (\text{mod} \; 2^{4}).$

The corresponding roots of congruence are $\{0, 4, 8, 12\}$ and $\{2, 6, 10, 14\},$ respectively. Therefore, $\overbrace{G}(2, 2^{4}) = 2K_{4}$ . There are three congruences for $\beta = 5$

$\eta^{2}\equiv 0\; (\text{mod} \; 2^{5}), \; \eta^{2}\equiv 4\; (\text{mod} \; 2^{5}), \; \; \text{and} \; \; \eta^{2}\equiv 16\; (\text{mod} \; 2^{5}).$

The solution sets of these congruences are $\{0, 8, 16, 24\}$ , $\{2, 6, 10, 14, 18, 22, 26, 30\}$ , and $\{4, 12, 20, 28\},$ respectively. Therefore, there are two copies of $K_{4}$ and one copy of $K_{8}$ . For $\beta = 6$ , we have

$\eta^{2}\equiv 0\; (\text{mod} \; 2^{6}), \; \; \eta^{2}\equiv 4\; (\text{mod} \; 2^{6}), \; \eta^{2}\equiv 16\; (\text{mod} \; 2^{6}), \; \; \text{and}\; \eta^{2}\equiv 36\; (\text{mod} \; 2^{6}).$

The zeros of these congruences are $\{0, 8, 16, 24, 32, 40, 48, 54\}$ , $\{2, 18, 34, 50\}\bigcup\{ 62, 46, 30, 14\},$ $\{4, 12, 20, 28, 36, 44, 52, 60\}$ , and $\{6, 22, 38, 54\}\bigcup\{ 58, 42, 26, 10\},$ respectively. For $\beta = 7$ , there are 7 different congruences. We have

$\begin{eqnarray} \eta^{2}&\equiv & (2t)^{2}\; (\text{mod} \; 2^{7}), \; \; t = 0, 1, 3, 4, 5, 7, \end{eqnarray}$

(2.1)

$\begin{eqnarray} \eta^{2}&\equiv & 16\; (\text{mod} \; 2^{7}). \end{eqnarray}$

(2.2)

Roots of these congruences are $\{ 2t+16m | m = 0, 1, 2\cdots 2^{3}-1\}$ , $\{ 2t+32m | m = 0, 1, 2\cdots 2^{2}-1\}\bigcup\{ 2^{7}-32m-2t| m = 0, 1, 2\cdots 2^{2}-1\}$ , $t = 1, 3, 4, 5, 7$ , $\{ 4+8m | m = 0, 1, 2\cdots 2^{4}-1\}$ . Hence, there are six copies of $K_{8}$ and one copy of $K_{16}$ . For $\beta = 8$ , there are 12 congruences. We have

$\begin{eqnarray} \eta^{2} &\equiv & (2+4t)^{2}\; (\text{mod} \; 2^{8}), \; \; t = 0, 1, \cdots , 2^{3}-1, \end{eqnarray}$

(2.3)

$\begin{eqnarray} \eta^{2}&\equiv & (4t)^{2}\; (\text{mod} \; 2^{8}), \; \; t = 0, 1, \cdots , 2^{2}-1. \end{eqnarray}$

(2.4)

Zeroes of these congruences are $\{ 2+4t+64m | m = 0, 1, 2\cdots 2^{2}-1\}\bigcup\{ 2^{8}-64m-4t-2| m = 0, 1, 2\cdots 2^{2}-1\}$ , $t = 0, 1, \cdots, 2^{3}-1$ , $\{ 4t+32m | m = 0, 1, 2\cdots 2^{4}-1\}$ , $\{4t+32m | m = 0, 1, 2\cdots 2^{3}-1\}\bigcup\{ 2^{8}-32m-4t| m = 0, 1, 2\cdots 2^{3}-1\}$ , $t = 1, \cdots, 2^{2}-1$ . Thus, we have $\overbrace{G}(2, 2^{8}) = 8K_{4}\bigoplus4K_{16}$ . For $\beta = 9$ , there are 23 different congruences. We have

$\begin{eqnarray} \eta^{2} &\equiv & (2+4t)^{2}\; (\text{mod} \; 2^{9}), \; \; t = 0, 1, \cdots , 2^{4}-1, \end{eqnarray}$

(2.5)

$\begin{eqnarray} \eta^{2}&\equiv & (4t)^{2}\; (\text{mod} \; 2^{9}), \; \; t = 0, 1, 3, 4, 5, 7, \end{eqnarray}$

(2.6)

$\begin{eqnarray} \eta^{2}&\equiv & 64\; (\text{mod} \; 2^{9}). \end{eqnarray}$

(2.7)

Solutions of congruences (2.5) $-$ (2.7) are $\{ 2+4t+128m | m = 0, 1, 2\cdots 2^{2}-1\}\bigcup\{ 2^{9}-128m-4t-2| m = 0, 1, 2\cdots 2^{2}-1\}$ , $t = 0, 1, \cdots, 2^{6}-1$ , $\{ 4t+32m | m = 0, 1, 2\cdots 2^{4}-1\}$ , $\{ 4t+64m | m = 0, 1, 2\cdots 2^{3}-1\}\bigcup\{ 2^{9}-64m-4t| m = 0, 1, 2\cdots 2^{3}-1\}$ , $t = 1, 3, 4, 5, 7$ , $\{ 8+16m | m = 0, 1, 2\cdots 2^{5}-1\}$ . That is, $\overbrace{G}(2, 2^{9}) = 16K_{8}\bigoplus 6K_{16}\bigoplus K_{32}$ . For $\beta = 10$ , there are 44 different congruences. We have

$\begin{eqnarray} \eta^{2} &\equiv & (2+4t)^{2}\; (\text{mod} \; 2^{10}), \; \; t = 0, 1, \cdots , 2^{5}-1, \end{eqnarray}$

(2.8)

$\begin{eqnarray} \eta^{2}&\equiv & (4+8t)^{2}\; (\text{mod} \; 2^{10}), \; \; t = 0, 1, \cdots , 2^{3}-1, \end{eqnarray}$

(2.9)

$\begin{eqnarray} \eta^{2}&\equiv & (8t)^{2}\; (\text{mod} \; 2^{10}), \; \; \; \; \; \; \; t = 0, 1, \cdots , 2^{2}-1, \end{eqnarray}$

(2.10)

Sequences of roots of congruences (2.8) $-$ (2.10) are $\{ 2+4t+256m | m = 0, 1, 2\cdots 2^{2}-1\}\bigcup\{ 2^{10}-256m-4t-2| m = 0, 1, 2\cdots 2^{2}-1\}$ $t = 0, 1, \cdots, 2^{5}-1$ , $\{ 4+8t+128m | m = 0, 1, 2\cdots 2^{3}-1\}\bigcup\{ 2^{10}-128n-8t-4| m = 0, 1, 2\cdots 2^{3}-1\}$ , $t = 0, 1, \cdots, 2^{3}-1$ , $\{ 8t+64m | m = 0, 1, 2\cdots 2^{5}-1\}$ , $\{ 8t+64m | m = 0, 1, 2\cdots 2^{4}-1\}\bigcup\{ 2^{10}-64m-8t| m = 0, 1, 2\cdots 2^{4}-1\}$ , $t = 1, \cdots, 2^{2}-1$ . That is, $\overbrace{G}(2, 2^{10}) = 32K_{8}\bigoplus 8K_{16}\bigoplus 4K_{32}$ . For $\beta = 11$ ,

$\begin{eqnarray} \eta^{2} &\equiv & (2+4t)^{2}\; (\text{mod} \; 2^{11}), \; \; t = 0, 1, \cdots , 2^{6}-1, \end{eqnarray}$

(2.11)

$\begin{eqnarray} \eta^{2}&\equiv & (4+8t)^{2}\; (\text{mod} \; 2^{11}), \; \; t = 0, 1, \cdots , 2^{4}-1, \end{eqnarray}$

(2.12)

$\begin{eqnarray} \eta^{2}&\equiv & (8t)^{2}\; (\text{mod} \; 2^{11}), \; \; \; \; \; \; \; t = 0, 1, 3, 4, 5, 7, \end{eqnarray}$

(2.13)

$\begin{eqnarray} \eta^{2}&\equiv & 256\; (\text{mod} \; 2^{11}). \end{eqnarray}$

(2.14)

Zeroes of congruences (2.11) $-$ (2.14) are $\{ 2+4t+512m | m = 0, 1, 2\cdots 2^{2}-1\}\bigcup\{ 2^{11}-512m-4t-2| m = 0, 1, 2\cdots 2^{2}-1\}$ , $t = 0, 1, \cdots, 2^{6}-1$ , $\{ 4+8t+256m | m = 0, 1, 2\cdots 2^{3}-1\}\bigcup\{ 2^{11}-256m-8t-4| m = 0, 1, 2\cdots 2^{3}-1\}$ , $t = 0, 1, \cdots, 2^{4}-1$ , $\{ 8t+64m | m = 0, 1, 2\cdots 2^{5}-1\}$ , $\{ 8t+128m | m = 0, 1, 2\cdots 2^{4}-1\}\bigcup\{ 2^{11}-128m-8t| m = 0, 1, 2\cdots 2^{4}-1\}$ , $t = 1, 3, 4, 5, 7$ , $\{ 16+32m | m = 0, 1, 2\cdots 2^{6}-1\}$ .

Therefore, we have $\overbrace{G}(2, 2^{11}) = 64K_{8}\bigoplus 16K_{16}\bigoplus 6K_{32}\bigoplus K_{64}$ . For $\beta = 12$ , there are 172 congruences. We have

$\begin{eqnarray} \eta^{2} &\equiv & (2+4t)^{2}\; (\text{mod} \; 2^{12}), \; \; t = 0, 1, \cdots , 2^{7}-1, \end{eqnarray}$

(2.15)

$\begin{eqnarray} \eta^{2}&\equiv & (4+8t)^{2}\; (\text{mod} \; 2^{12}), \; \; t = 0, 1, \cdots , 2^{5}-1, \end{eqnarray}$

(2.16)

$\begin{eqnarray} \eta^{2}&\equiv & (8+16t)^{2}\; (\text{mod} \; 2^{12}), \; \; t = 0, 1, \cdots , 2^{3}-1, \end{eqnarray}$

(2.17)

$\begin{eqnarray} \eta^{2}&\equiv & (16t)^{2}\; (\text{mod} \; 2^{12}), \; \; t = 0, 1, \cdots , 2^{2}-1. \end{eqnarray}$

(2.18)

Sets of roots of congruences (2.15) $-$ (2.18) are $\{ 2+4t+1024m | m = 0, 1, 2\cdots 2^{2}-1\}\bigcup\{ 2^{12}-1024m-4t-2| m = 0, 1, 2\cdots 2^{2}-1\}$ , $t = 0, 1, \cdots, 2^{7}-1$ , $\{ 4+8t+512m | m = 0, 1, 2\cdots 2^{3}-1\}\bigcup\{ 2^{12}-512m-8t-4| m = 0, 1, 2\cdots 2^{3}-1\}$ , $t = 0, 1, \cdots, 2^{5}-1$ , $\{ 8t+256m | m = 0, 1, 2\cdots 2^{4}-1\}\bigcup\{ 2^{12}-256m-8t| m = 0, 1, 2\cdots 2^{4}-1\}$ , $t = 0, 1, \cdots, 2^{3}-1$ , $\{ 16t+128m | m = 0, 1, 2\cdots 2^{6}-1\}$ , $\{ 16t+128m | m = 0, 1, 2\cdots 2^{5}-1\}\bigcup\{ 2^{12}-128m-8t| m = 0, 1, 2\cdots 2^{5}-1\}$ , $t = 1, \cdots, 2^{2}-1$ . That is, $\overbrace{G}(2, 2^{12}) = 128K_{8}\bigoplus 32K_{16}\bigoplus 8K_{32}\bigoplus 4K_{64}$ .

The generalize sequence for $\beta \geq 9,$ where $\beta$ is an odd number, there are $\frac{2^{4}(2^{\beta-7}-1)+21}{3}$ number of congruences. we have

$\begin{eqnarray} \eta^{2} &\equiv & (2^{i}+2^{i+1}t)^{2}\; (\text{mod} \; 2^{\beta}), \; \; t = 0, 1, \cdots , 2^{\beta-2i-3}-1, \end{eqnarray}$

(2.19)

$\begin{eqnarray} & &\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; i = 1, 2, 3, \cdots, \frac{\beta-7}{2}, \\ \eta^{2}&\equiv & (2^{\frac{\beta-5}{2}}t)^{2}\; (\text{mod} \; 2^{\beta}), \; \; \; \; \; \; \; t = 0, 1, 3, 4, 5, 7, \end{eqnarray}$

(2.20)

$\begin{eqnarray} \eta^{2}&\equiv & 2^{\beta-3}\; (\text{mod} \; 2^{\beta}). \end{eqnarray}$

(2.21)

Sequences of roots of congruences (2.19) $-$ (2.21) are $\{ 2^{i}+2^{i+1}t+2^{\beta-i-1}m | m = 0, 1, 2\cdots 2^{i+1}-1\}\bigcup\{ 2^{\beta} -2^{\beta-i-1}m-2^{i+1}t-2^{i}| m = 0, 1, 2\cdots 2^{i+1}-1\}$ , $t = 0, 1, \cdots, 2^{\beta-2i-3}-1$ , $i = 1, 2, 3, \cdots, \frac{\beta-7}{2}$ , $\{ 2^{\frac{\beta-5}{2}}t+2^{\frac{\beta+1}{2}}m | m = 0, 1, 2, \cdots 2^{\frac{\beta-1}{2}}-1\},$ $\{ 2^{\frac{\beta-5}{2}}t+2^{\frac{\beta+3}{2}}m | m = 0, 1, 2, \cdots 2^{\frac{\beta-3}{2}}-1\} \bigcup\{ 2^{\beta}-2^{\frac{\beta+3}{2}}m-2^{\frac{\beta-5}{2}}k| m = 0, 1, 2, \cdots 2^{\frac{\beta-3}{2}}-1\}$ , $t = 1, 3, 4, 5, 7$ , $\{2^{\frac{\beta-3}{2}}+2^{\frac{\beta-1}{2}}m | m = 0, 1, 2\cdots 2^{\frac{\beta+1}{2}}-1\}$ . That is, $\overbrace{G}(2, 2^{\beta}) = \bigoplus\limits_{i = 1}^{\frac{\beta -7}{2}}2^{\beta-5-2(i-1)}K_{2^{3+(i-1)}} ~~~\bigoplus 6K_{2^{(\beta-1)/2}}~~ \bigoplus K_{2^{(\beta+1)/2}}$ .

For second case when $\beta$ is an even number and $\beta\ge10$ , we have $\frac{2^{5}(2^{\beta-8}-1)+36}{3}$ congruences as follows

$\begin{eqnarray} \eta^{2} &\equiv & (2^{i}+2^{i+1}t)^{2}\; (\text{mod} \; 2^{\beta}), \; \; t = 0, 1, \cdots , 2^{\beta-2i-3}-1, \end{eqnarray}$

(2.22)

$\begin{eqnarray} & &\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; i = 1, 2, 3, \cdots, \frac{\beta-6}{2}, \\ \eta^{2}&\equiv & (2^{\frac{\beta-4}{2}}t)^{2}\; (\text{mod} \; 2^{\beta}), \; \; \; \; \; \; \; t = 0, 1, \cdots, 2^{2}-1. \end{eqnarray}$

(2.23)

Sequences of zeroes of congruences (2.22) $-$ (2.23) are $\{ 2^{i}+2^{i+1}t+2^{\beta-i-1}m | m = 0, 1, 2\cdots 2^{i+1}-1\}\bigcup\{ 2^{\beta}-2^{\beta-i-1}m-2^{i+1}t-2^{i}| m = 0, 1, 2\cdots 2^{i+1}-1\}$ , $t = 0, 1, \cdots, 2^{\beta-2i-3}-1$ , $i = 1, 2, 3, \cdots, \frac{\beta-6}{2}$ , $\{ 2^{\frac{\beta-4}{2}}t+2^{\frac{\beta+2}{2}}m | m = 0, 1, 2\cdots, 2^{\frac{\beta}{2}}-1\}$ , $\{ 2^{\frac{\beta-4}{2}}t+2^{\frac{\beta+2}{2}}m | m = 0, 1, 2\cdots, 2^{\frac{\beta-2}{2}}-1\} \bigcup\{ 2^{\beta}-2^{\frac{\beta+2}{2}}m-2^{\frac{\beta-4}{2}}t| m = 0, 1, 2^{\frac{\beta-2}{2}}-1\}$ , $t = 1, \cdots, 2^{2}-1$ .

Thus, $\overbrace{G}(2, 2^{\beta}) = \bigoplus\limits_{i = 1}^{\frac{\beta -6}{2}}2^{\beta-2i-3}K_{2^{2+i}} \bigoplus 2^{2}K_{2^{\beta/2}}$ .

Now, we discuss the unit elements of $Z_{m}.$ For $m = 2, 4$ , the result is straightforward. For $\beta\ge3$ , the graph $\overbrace{G}(2, m)$ contains $\phi(2^{\beta}) = 2^{\beta-1}~~$ number of vertices. We calculate the least positive residues of the square of the integers, which are smaller than and relatively prime with $m.$ Hence, there are $\phi(2^{\beta}) = 2^{\beta-1}~~$ obtained. By Theorem 1.2, the congruence $x^{2}\equiv a \; (\text{mod}\; 2^{\beta})$ has either no solution or exactly four incongruent solutions. This implies that, there are always $\frac{\phi(m)}{4} = \frac{\phi(2^{\beta-1}~~)}{4} = 2^{\beta-3}$ quadratic residues among all the vertices. Thus $\overbrace{G}(2, m) = 2^{\beta-3}K_{4}$ . By combining both cases, we get the desired result. The quadratic residues graph for $n = 128$ is shown in Figure 2.

Figure 2.

$\overbrace{G}(2, 2^{7}) = 6K_{8}\bigoplus K_{16}\bigoplus 16 K_{4}$ .

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Theorem 2.2. Let $q$ be an odd prime. Then $\overbrace{G}(2, q^{\beta})$

$= \left\{ \begin{array}{ll} K_{1} \bigoplus \frac{q-1}{2}K_{2}, & {\rm\; if\; }\beta = 1, \\ K_{q}\bigoplus \frac{q(q-1)}{2}K_{2}, & {\rm\; if\; }\beta = 2, \\ \Big{(}\frac{q-1}{2}\Big{)}K_{2p}\bigoplus K_{q}\bigoplus \frac{q^{2}(q-1)}{2}K_{2}, & {\rm\; if\; }\beta = 3, \\ \Big{(}\frac{q-1}{2}\Big{)}qK_{2q}\bigoplus K_{q^{2}}\bigoplus \frac{q^{3}(q-1)}{2}K_{2}, & {\rm\; if\; }\beta = 4, \\ \bigoplus\limits_{i = 1}^{\frac{\beta -1}{2}}\Big{(}\frac{q^{\beta-2i}-q^{\beta-2i-1}}{2}\Big{)}K_{2q^{i}}\bigoplus K_{q^{(\beta -1)/2}}\bigoplus \frac{q^{\beta-1}~~(q-1)}{2}K_{2}, & {\rm\; if\; }\beta\geq 5, \\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; {\rm\; and\; } \beta\equiv 1\; (\text{mod}\; 2), \\ \bigoplus\limits_{i = 1}^{\frac{\beta -2}{2}}\Big{(}\frac{q^{\beta-2i}-q^{\beta-2i-1}}{2}\Big{)}K_{2q^{i}}\bigoplus K_{q^{\beta/2}}\bigoplus \frac{q^{\beta-1}~~(q-1)}{2}K_{2}, & {\rm\; if\; }\beta\geq 6, \\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; {\rm\; and\; } \beta\equiv 0\; (\text{mod}\; 2).\\ \end{array} \right.$

Proof. To prove this theorem first we assume zero-divisors of the ring $Z_{m}$ . Let $q$ is an odd prime and $S = \{ tq^{\beta} | t = 0, 1, 2, \cdots, q^{\beta-1}~~-1 \}$ be zero-divisors including zero of $q^{\beta}$ for each positive integer $\beta$ . To solve congruence $\eta^{2}\equiv \beta^{2} \; (\text{mod}\; q^{\beta})$ for each $\beta\geq1$ , we start with $\eta^{2}\equiv \beta^{2} \; (\text{mod}\; q)$ . In this case, $0$ is only root of this congruence, but there is no edge between two vertices when they are same, so $\overbrace{G}(2, q) = K_{1}$ . For $\beta = 2$ , there is only one congruence namely $\eta^{2}\equiv 0 \; (\text{mod}\; q^{2})$ , roots of this congruence are $\eta = 0, q, 2q, \cdots, (q-1)q$ . There are $q$ solutions and will be complete graph of order $q$ . For $\beta = 3,$ there are $\frac{q+1}{2}$ congruences. We have

$\begin{eqnarray} \eta^{2} &\equiv & 0\; (\text{mod} \; q^{3}), \end{eqnarray}$

(2.24)

$\begin{eqnarray} \eta^{2}&\equiv & (qt)^{2}\; (\text{mod} \; q^{3}), \; \; t = 1, 2, \cdots, \frac{q-1}{2}. \end{eqnarray}$

(2.25)

Zeros of these congruences are $\{\eta = q^{2}(m-1) | m = 1, 2, \cdots, q \}$ , and $\{\eta = q^{2}(m-1)+qt |\; t = 1, 2, \cdots, \frac{q-1}{2}, \; \; m = 1, 2, \cdots, q \}$ $\bigcup \{\eta = q^{3}-q^{2}(m-1)-qt |\; t = 1, 2, \cdots, \frac{q-1}{2}, \; \; m = 1, 2, \cdots, q \},$ respectively. For $\beta = 4$ , the number of distinct congruences is $(q^{2} - q + 1)/2.$ These are

$\begin{eqnarray} \eta^{2} &\equiv & 0\; (\text{mod} \; q^{4}), \end{eqnarray}$

(2.26)

$\begin{array}{l}\eta^{2} \equiv (qt)^{2}\; (\text{mod} \; q^{4}), \; t = 1, 2, \cdots, \frac{q^{2}-1}{2}, \\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q-1}{2}. \end{array}$

(2.27)

Sequences of roots of these congruences are $\{\eta = q^{2}(m-1) | m = 1, 2, \cdots, q^{2} \}$ , and $\{\eta = q^{3}(m-1)+qt |\; t = 1, 2, \cdots, \frac{q^{2}-1}{2}, {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q-1}{2}, \; \; m = 1, 2, \cdots, q \}$ $\bigcup \{\eta = q^{4}-q^{3}(-1)-pt |\; t = 1, 2, \cdots, \frac{p^{2}-1}{2}, {\rm\; but\; } t\neq pl, \; \; l = 1, 2, \cdots, \frac{q-1}{2}, \; \; m = 1, 2, \cdots, q \},$ respectively.

For $\beta = 5$ , we have

$\begin{eqnarray} \eta^{2} &\equiv & 0\; (\text{mod} \; q^{5}), \end{eqnarray}$

(2.28)

$\begin{eqnarray} \eta^{2}&\equiv & (q^{2}t)^{2}\; (\text{mod} \; q^{5}), \; \; t = 1, 2, \cdots, \frac{q-1}{2} \end{eqnarray}$

(2.29)

$\begin{array}{l} \eta^{2} \equiv (qt)^{2}\; (\text{mod} \; q^{5}), \; t = 1, 2, \cdots, \frac{q^{3}-1}{2},\\ {\rm\; but\; } t\neq pl, \; \; l = 1, 2, \cdots, \frac{q^{2}-1}{2}. \end{array}$

(2.30)

Zeroes of these congruences are $\{\eta = q^{3}(m-1) | m = 1, 2, \cdots, q^{2} \}$ , $\{\eta = q^{3}(m-1)+q^{2}t |\; t = 1, 2, \cdots, \frac{q-1}{2}, \; \; m = 1, 2, \cdots, q^{2} \}$ $\bigcup \{\eta = q^{5}-q^{3}(m-1)-q^{2}t |\; t = 1, 2, \cdots, \frac{q-1}{2}, \; \; m = 1, 2, \cdots, q^{2} \}$ , and $\{\eta = q^{4}(m-1)+qt |\; t = 1, 2, \cdots, \frac{q^{3}-1}{2}, {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q^{2}-1}{2}, \; \; m = 1, 2, \cdots, q \}$ $\bigcup \{\eta = q^{5}-q^{4}(m-1)-qt |\; t = 1, 2, \cdots, \frac{q^{3}-1}{2}, {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q^{2}-1}{2}, \; \; m = 1, 2, \cdots, q \},$ respectively.

If $\beta = 6$ , then we have

$\begin{eqnarray} \eta^{2} &\equiv & 0\; (\text{mod} \; q^{6}), \end{eqnarray}$

(2.31)

$\begin{eqnarray} \eta^{2}&\equiv & (q^{2}t)^{2}\; (\text{mod} \; q^{6}), \; t = 1, 2, \cdots, \frac{q^{2}-1}{2}, \end{eqnarray}$

(2.32)

$\begin{array}{l} {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q-1}{2}\\ \eta^{2} \equiv (qt)^{2}\; (\text{mod} \; q^{6}), \; t = 1, 2, \cdots, \frac{q^{4}-1}{2},\\ \; \; \; \; {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q^{3}-1}{2}. \end{array}$

(2.33)

Sequences of roots of congruences (2.31) $-$ (2.33) are $\{\eta = q^{3}(m-1) | m = 1, 2, \cdots, q^{3} \}$ , $\{\eta = q^{4}(m-1)+q^{2}t |\; t = 1, 2, \cdots, \frac{q^{2}-1}{2}, \; {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q-1}{2}, \; \; m = 1, 2, \cdots, q^{2} \}$ $\bigcup \{\eta = q^{6}-q^{3}(m-1)-q^{2}t |\; t = 1, 2, \cdots, \frac{q^{2}-1}{2}, \; {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q-1}{2}, \; \; m = 1, 2, \cdots, q^{2} \}$ , and $\{\eta = q^{5}(m-1)+qt |\; t = 1, 2, \cdots, \frac{q^{4}-1}{2}, {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q^{3}-1}{2}, \; \; m = 1, 2, \cdots, q \}$ $\bigcup \{\eta = q^{6}-q^{5}(m-1)-qt |\; t = 1, 2, \cdots, \frac{q^{4}-1}{2}, {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q^{3}-1}{2}, \; \; m = 1, 2, \cdots, q \},$ respectively.

Now we are going to derive generalize sequence for both odd and even distinct congruences for $\beta\geq5$ and $\beta \equiv 1\; (\text{mod}\; 2)$ . These are

$\begin{eqnarray} \eta^{2}&\equiv & (q^{i}t)^{2}\; (\text{mod} \; q^{\beta}), \; t = 1, 2, \cdots, \frac{q^{\beta-2i}-1}{2}, \end{eqnarray}$

(2.34)

$\begin{eqnarray} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; {\rm\; but}& & t\neq ql, \; \; l = 1, 2, \cdots, \frac{q^{\beta-2i-1}-1}{2}, \; \; i = 1, 2, \cdots, \frac{\beta-1}{2}, \\ \eta^{2}&\equiv & 0\; (\text{mod} \; q^{\beta}). \end{eqnarray}$

(2.35)

Sequences of roots are $\bigcup\limits _{i = 1}^{\frac{\beta-1}{2}}{\{}\{\eta = q^{\beta-i}(m-1)+q^{i}t |\; t = 1, 2, \cdots, \frac{q^{\beta-2i}-1}{2}, {\rm\; but\; } l\neq ql, \; \; l = 1, 2, 3, \cdots, \frac{q^{\beta-2i-1}-1}{2}, m = 1, 2, \cdots, q^{i} \}$ , $\bigcup \{\eta = q^{\beta}-q^{\beta-i}(m-1)-q^{i}t |\; t = 1, 2, \cdots, \frac{q^{\beta-2i}-1}{2}, {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q^{\beta-2i-1}-1}{2}, \; \; m = 1, 2, \cdots, q^{i} \} \Big{\}}$ and $\{\eta = q^{\frac{\beta+1}{2}}(m-1) | m = 1, 2, \cdots, q^{\frac{\beta-1}{2}} \},$ respectively. Therefore, for every positive integer $\beta \geq 5$ with $\beta\equiv 1\; (\text{mod} \; 2)$ , $\overbrace{G}(2, q^{\beta}) = \bigoplus\limits_{i = 1}^{\frac{\beta -1}{2}}\Big{(}\frac{q^{\beta-2i}-q^{\beta-2i-1}}{2}\Big{)}K_{2q^{i}}\bigoplus K_{q^{(\beta -1)/2}}$ . In second case, when $\beta\geq6$ and $\beta \equiv 1\; (\text{mod}\; 2)$ , the number of distinct congruences is $\frac{q^{\beta-1}~~+q+2}{2(q+1)}$ . We have

$\begin{eqnarray} \eta^{2}&\equiv & (q^{i}t)^{2}\; (\text{mod} \; q^{\beta}), \; t = 1, 2, \cdots, \frac{q^{\beta-2i}-1}{2}, \end{eqnarray}$

(2.36)

$\begin{eqnarray} {\rm\; but}&&t\neq ql, \; \; l = 1, 2, \cdots, \frac{q^{\beta-2i-1}-1}{2}, \; \; i = 1, 2, \cdots, \frac{\beta-2}{2}, \\ \eta^{2} &\equiv & 0\; (\text{mod} \; q^{\beta}). \end{eqnarray}$

(2.37)

Zeroes of congruences $(2.36)$ and $(2.37)$ are $\bigcup\limits _{i = 1}^{\frac{\beta-2}{2}}{\{}\{\eta = q^{\beta-i}(m-1)+q^{i}t | \; t = 1, 2, \cdots, \frac{q^{\beta-2i}-1}{2}, {\rm\; but\; } t\neq ql, \; \; l = 1, 2, 3, \cdots, \frac{q^{\beta-2i-1}-1}{2}, \; m = 1, 2, \cdots, q^{i} \}$ $\bigcup \{\eta = q^{\beta}-q^{\beta-i}(m-1)-q^{i}t |\; t = 1, 2, \cdots, \frac{q^{\beta-2i}-1}{2}, {\rm\; but\; } t\neq ql, \; \; l = 1, 2, \cdots, \frac{q^{\beta-2i-1}-1}{2}, \; \; m = 1, 2, \cdots, q^{i} \} \Big{\}}$ , and $\{\eta = q^{\frac{\beta}{2}}(m-1) | m = 1, 2, \cdots, q^{\frac{\beta}{2}} \}$ , respectively. Thus, for $\beta\equiv6$ with $\beta\equiv 0\; (\text{mod} \; 2)$ , $\overbrace{G}(2, q^{\beta}) = \bigoplus\limits_{i = 1}^{\frac{\beta -2}{2}}\Big{(}\frac{q^{\beta-2i}-q^{\beta-2i-1}}{2}\Big{)}K_{2q^{i}}\bigoplus K_{q^{\beta/2}}.$

Now, we assume the set of unit elements of $Z_{m}.$ The graph $\overbrace{G}(2, m)$ contains $\phi(q^{\beta}) = q^{\beta-1}~~(q-1),$ vertices, where $\beta\geq1$ . We determine the least positive residue of the square of the integers which are less than and relatively prime with $m.$ Because, there are $\phi(q^{\beta}) = q^{\beta-1}~~(q-1)$ squares to be found. By Theorem 1.1, the congruence $x^{2}\equiv a \; (\text{mod}\; q^{\beta})$ has either no solution or exactly two incongruent solutions. This implies, there are always $\frac{\phi(q^{\beta})}{2} = \frac{(q^{\beta-1}~~(q-1))}{2}$ quadratic residues among all the vertices. Thus, $\overbrace{G}(2, q^{\beta}) = \frac{(q^{\beta-1}~~(q-1))}{2}K_{2}$ . By combining both cases, we get the desired result. Quadratic graph for $m = 162$ is shown in Figure 3.

Figure 3.

$\overbrace{G}(2, 243) = 9K_{6}\bigoplus K_{9}\bigoplus K_{18}\bigoplus 81 K_{2}$ .

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3. Quadratic residues graphs over $Z_{2^{\beta}q},$ $Z_{qp}$ and $Z_{2q^{\beta}}$

In this section, we characterize quadratic residues graphs for $n = 2^{\beta}q, 2q^{\beta}, qp$ .

Theorem 3.1. Let $q$ be an odd prime. Then $\overbrace{G}(2, 2^{\beta}q)$

$= \left\{ \begin{array}{ll} 2K_{1}\bigoplus (q-1)K_{2}, & {\rm\; if\; }\beta = 1, \\ 2K_{2} \bigoplus(q-1)K_{4}, & {\rm\; if\; }\beta = 2, \\ 2K_{2} \bigoplus qK_{4}\bigoplus\frac{q-1}{2}K_{8}, & {\rm\; if\; }\beta = 3, \\ 4K_{4} \bigoplus 2(q-1)K_{8}, & {\rm\; if\; }\beta = 4, \\ 6K_{4} \bigoplus(3q-2)K_{8}\bigoplus\frac{q-1}{2}K_{16}, & {\rm\; if\; }\beta = 5, \\ 8K_{4} \bigoplus 4qK_{8}\bigoplus 2(q-1)K_{16}, & {\rm\; if\; }\beta = 6, \\ 16K_{4} \bigoplus 2(4q-1)K_{8}\bigoplus(3q-2)K_{16}\bigoplus\frac{q-1}{2}K_{32}, & {\rm\; if\; }\beta = 7, \\ 32K_{4} \bigoplus 8(2q-1)K_{8}\bigoplus 4qK_{16}\bigoplus 2(q-1)K_{32}, & {\rm\; if\; }\beta = 8, \\ 64K_{4} \bigoplus 16(2q-1)K_{8}\bigoplus(8q-2)K_{16}\bigoplus (3q-2)K_{32}\bigoplus\frac{q-1}{2}K_{64}, & {\rm\; if\; }\beta = 9, \\ \bigoplus\limits_{i = 1}^{2}2^{\beta-2i-1}K_{2^{i+1}} \bigoplus\limits_{i = 1}^{\frac{\beta-8}{2}} \Big{(}2^{\beta-4-2i}q-2^{\beta-5-2i}\Big{)}K_{2^{3+i}}\bigoplus\\ 4qK_{2^{\frac{\beta}{2}}} \bigoplus 2(q-1) K_{2^{\frac{\beta+2}{2}}}\bigoplus\frac{2^{\beta-1}~~(q-1)}{8}K_{8} , & {\rm\; if\; }\beta\geq 10, \\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; {\rm\; and\; } \beta\equiv 0\; (\text{mod}\; 2), \\ \bigoplus\limits_{i = 1}^{2}2^{\beta-2i-1}K_{2^{i+1}} \bigoplus\limits_{i = 1}^{\frac{\beta-9}{2}} \Big{(}2^{\beta-4-2i}q-2^{\beta-5-2i}\Big{)}K_{2^{3+i}}\bigoplus\\ (8q-2)K_{2^{\frac{\beta-1}{2}}} \bigoplus (3q-2) K_{2^{\frac{\beta+1}{2}}}\bigoplus\frac{q-1}{2} K_{2^{\frac{\beta+3}{2}}}\bigoplus\frac{2^{\beta-1}~~(q-1)}{8}K_{8} , & {\rm\; if\; }\beta\geq 11, \\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; {\rm\; and\; } \beta\equiv 1\; (\text{mod}\; 2).\\ \end{array} \right.$

Proof. Let $n = 2^{\beta}q$ be an integer, where $q$ is an odd prime. For $\beta = 1$ , $S = \{ 2m|\; m = 0, 1, 2, \cdots q-1\}\bigcup\{q\}$ is set of zero-divisors of $Z_{2q}$ including $0$ . There are $\frac{q+1}{2}$ distinct congruences. We have

$\eta^{2} \equiv 0\; (\text{mod} \; 2q), \; \eta^{2} \equiv q\; (\text{mod} \; 2q), \; \text{and}\; \eta^{2}\equiv (2t)^{2}\; (\text{mod} \; 2q), \; \; \; t = 1, 2, \cdots, \frac{q-1}{2},$

$\gamma = 0$ , $\gamma = q$ , and $\gamma = 2t, \; 2q-2t, \; \; t = 1, 2, 3, \cdots, \frac{q-1}{2}$ zeroes of congruences, respectively. When $\beta = 2$ , there exit following congruences given as

$\eta^{2} \equiv 0\; (\text{mod} \; 2^{2}q), \; \eta^{2} \equiv q\; (\text{mod} \; 2^{2}q), \; \text{and}\; \eta^{2}\equiv (2t)^{2}\; (\text{mod} \; 2^{2}q), \; \; \; t = 1, 2, \cdots, \frac{q-1}{2}.$

Sets of roots of these congruence are $\{ 0, 2q \}$ , $\{ q, 3q \}$ , $\{ 2t, 2q-2t\}\bigcup\{2^{2}q-2t, 2^{2}q-2q+2t\}, \; \; \; t = 1, 2, \cdots, \frac{q-1}{2},$ respectively. There are $q+2$ congruences for $\beta = 3$ . They are $\eta^{2} \equiv 0\; (\text{mod} \; 2^{3}q), \; \eta^{2} \equiv (2 q)^{2}\; (\text{mod} \; 2^{3}q), \; \; \eta^{2} \equiv q^{2}\; (\text{mod} \; 2^{3}q), \; \; \text{and}\; \eta^{2}\equiv (2t)^{2}\; (\text{mod} \; 2^{3}q), \; \; \;$ $t = 1, 2, \cdots, q-1.$ The zeroes of these congruences are $\{ 0, 2^{2}q \}$ , $\{ 2q, 2^{3}q-2q \}$ , $\{ q, 3q, 5q, 7q \}$ and $\{ 2t, 2^{2}q-2t\}\bigcup\{2^{3}q-2t, 2^{3}q-2^{2}q+2t\}, \; \; \; t = 1, 2, \cdots, q-1$ , respectively. For $\beta = 4$ , there are $q + 3$ congruences. We have

$\eta^{2} \equiv 0\; (\text{mod} \; 2^{4}q), \; \eta^{2} \equiv q^{2}\; (\text{mod} \; 2^{4}q), \; \eta^{2} \equiv (2q)^{2}\; (\text{mod} \; 2^{4}q), \; \; \eta^{2} \equiv (3q)^{2}\; (\text{mod} \; 2^{4}q),$

${and\; \; \; \; }\eta^{2}\equiv (2t)^{2}\; (\text{mod} \; 2^{4}q), \; \; \; t = 1, 2, \cdots, q-1,$

zeroes are $\{ 0, 2^{2}q, 2^{3}q, 2^{2}q+ 2^{3}q \}$ , $\{ q, 7p, 9p, 15q\}$ , $\{ 2q, 6q, 10q, 14q\}$ , $\{ 3q, 5q, 11q, 13q\}$ and $\{ 2t, 2^{2}q-2t, 2^{2}q+2t, 2^{3}q-2t \}\bigcup\{2^{4}q-2t, 2^{4}q-2^{2}q+2t, 2^{4}q-2^{2}q-2t, 2^{4}q-2^{3}q+2t \}, \; \; \;$ $t = 1, 2, \cdots, q-1$ , respectively.

For $\beta = 5$ , there are $(3q+11)/2$ congruences

$\begin{eqnarray} \eta^{2} &\equiv& 0\; (\text{mod} \; 2^{5}q), \end{eqnarray}$

(3.1)

$\begin{eqnarray} \eta^{2} & \equiv& (qt)^{2}\; (\text{mod} \; 2^{5}q), t = 1, 3, 5, 7, \end{eqnarray}$

(3.2)

$\begin{eqnarray} \eta^{2} &\equiv& (4q)^{2}\; (\text{mod} \; 2^{5}q), \end{eqnarray}$

(3.3)

$\begin{eqnarray} \eta^{2} &\equiv& (2q)^{2}\; (\text{mod} \; 2^{5}q), \end{eqnarray}$

(3.4)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{5}q), \; \; t = 2i, \; i = 1, 2, 3, \cdots, q-1, \end{eqnarray}$

(3.5)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{5}q), \; \; t = 2i-1, \; i = 1, 2, 3, \cdots, \frac{q-1}{2}. \end{eqnarray}$

(3.6)

Zeroes of congruences (3.1) $-$ (3.6) are $\{0, 2^{3}q, 2^{4}q, 2^{3}q+2^{4}q \}$ , $\bigcup \limits_{t\in \{1, 3, 5, 7\}} \{ qt, 2^{4}q+qt \}\bigcup \{2^{5}q-qt, 2^{5}q-2^{4}q- qt\}$ , $\{4q, 12q, 20q, 28q \}$ , $\{2q, 6q, 10q, 14q, 18q, 22q, 26q, 30q \}$ , $\bigcup_{\substack{t = 2l, \\1 \leq l \leq q-1}}\Big{\{}\{ 2^{3} q j + 2t, \; \; j = 0, 1, 2, \cdots, 2^{2}-1 \}\cup \{ 2^{5}q-2^{3} q j - 2t, \; \; j = 0, 1, 2, \cdots, 2^{2}-1\} \Big{\}},$ $\bigcup \limits_{ l = i}^ {\frac{q-1}{2} }\Big{\{}\{ 2^{2} q i + 4l-2, \; \; i = 0, 1, 2, \cdots, 2^{3}-1 \}\cup \{ 2^{5}q-2^{2} q i - 4l+2, \; \; i = 0, 1, 2, \cdots, 2^{3}-1\} \Big{\}}.$

For $\beta = 6$ , we have

$\begin{eqnarray} \eta^{2} &\equiv& (qt)^{2}\; (\text{mod} \; 2^{6}q), \; \; t = 2i-1, \; i = 1, 2, 3, \cdots, 2^{3}, \end{eqnarray}$

(3.7)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{6}q), \; \; t = qi, \; i = 0, 1, 2, 3, \end{eqnarray}$

(3.8)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{6}q), \; \; t = 4l, \; \; l = 1, 2 \cdots \frac{q-1}{2} , \end{eqnarray}$

(3.9)

$\begin{eqnarray} \eta^{2} &\equiv& (2)^{2}\; (\text{mod} \; 2^{6}q), \; \; t = 3q-4i-qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{3q}{4} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 2, \cdots, \lfloor \frac{3q-4i}{q} \rfloor. \end{eqnarray}$

(3.10)

Solution sets of congruences (3.7) $-$ (3.10) are

$\begin{eqnarray*} \quad \quad \quad \quad \; \; \bigcup \limits_{t = 2i-1, i = 1}^{\; 2^{3}} &&\{ qt, 2^{5}q+qt \}\bigcup \{2^{6}q-qt, 2^{6}q-2^{5}q-qt\}, \\ && \{ 0, 2^{3}q, 2^{4}q, 2^{5}q, 2^{3}q+2^{4}q, 2^{3}q+2^{5}q, 2^{4}q+2^{5}q, 2^{3}q+2^{4}q+ 2^{5}q \}, \end{eqnarray*}$

$\begin{eqnarray*} \bigcup \limits_{t = qi, i = 1}^{3} && \{ 2t, 2^{4}t-2t, 2^{4}t+2t, 2^{5}t-2t \}\bigcup \{ 2^{6}q-2t, 2^{6}q-2^{4}t\\&+&2t, 2^{6}q-2^{4}t-2t, 2^{6}q-2^{5}t+2t\}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \quad \quad \quad \; \bigcup \limits_{t = 4l, l = 1}^{\frac{q-1}{2}}&& \{ 2^{3}qj+2t, \; j = 0, 1, 2, \cdots, 2^{3}-1 \}\bigcup \{ 2^{6}q-2^{3}qj-2t, \; j = 0, 1, \\&& 2, \cdots, 2^{3}-1\}, \end{eqnarray*}$

$\bigcup \limits_{1\leq i\leq \lfloor \frac{3q}{4} \rfloor, j = 0}^{\lfloor \frac{3q-4i}{q}\rfloor} \{ 2^{4}ql+6p-8i-2qj, \; l = 0, 1, \cdots, 2^{2}-1 \}$ $\bigcup \{ 2^{6}q-2^{4}ql-6q+8i+2qj, \; l = 0, 1, \cdots, 2^{2}-1\}$ $\bigcup \{ 2^{4}ql+ 2^{4}+6q-8i-2qj, \; l = 0, 1, \cdots, 2^{2}-1 \}\bigcup \{ 2^{6}q-2^{4}ql- 2^{4}-6q+8i+2qj, \; l = 0, 1, \cdots, 2^{2}-1\}.$

For $\beta = 7$ , we obtain

$\begin{eqnarray} \eta^{2} &\equiv& (qt)^{2}\; (\text{mod} \; 2^{7}q), \; \; t = 2i-1, \; i = 1, 2, 3, \cdots, 2^{4}, \end{eqnarray}$

(3.11)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{7}q), \; \; t = qi, \; i = 0, 1, 3, 4, 5, 7, \end{eqnarray}$

(3.12)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{7}q), \; \; t = 2q, \end{eqnarray}$

(3.13)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{7}q), \; \; t = 4l, \; \; l = 1, 2, \cdots, q-1, \end{eqnarray}$

(3.14)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{7}q), \; \; t = 7q-8i-2qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{7q}{8} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 2, \cdots, \lfloor \frac{7q-8i}{2q} \rfloor, \end{eqnarray}$

(3.15)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{7}q), \; \; t = 4l-2, \; \; l = 1, 2 \cdots \frac{q-1}{2}. \end{eqnarray}$

(3.16)

Zeroes of congruences (3.11) $-$ (3.16) are

$\begin{eqnarray*} \quad \bigcup \limits_{t = 2i-1, i = 1}^{\; 2^{4}} && \{ qt, 2^{6}q+qt \}\bigcup \{2^{7}q-qt, 2^{7}q-2^{6}q-qt\}, \\ &&\{ 0, 2^{4}q, 2^{5}q, 2^{6}q, 2^{4}q+2^{5}q, 2^{4}q+2^{6}q, 2^{5}q+2^{6}q, 2^{4}q+2^{5}q+ 2^{6}q \}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \quad \bigcup\limits_{\substack{t = qi, \\i \in \{ 1, 3, 4, 5, 7 \}}}&&\Big{\{}\{2t, 2^{5}t-2t, 2^{5}t+2t, 2^{6}t-2t \}\cup \{ 2^{7}q-2t, 2^{7}q-2^{5}t+2t, 2^{7}q-2^{5}t\\&&-2t, 2^{7}q-2^{6}t+2t\} \Big{\}}, \{2^{3}qj+2t, t = 2q, \; j = 0, 1, 2, \cdots, 2^{4}-1 \}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \quad\; \; \; \bigcup\limits_{\substack{t = 4l, \\ 1 \leq l \leq q-1}}\; \; &&\Big{\{}\{ 2^{4} q j + 2t, \; \; j = 0, 1, 2 , \cdots, 2^{3}-1 \}\cup \{ 2^{7}q-2^{4} q j - 2t, \; \; j = 0, 1, 2 , \\ && \cdots, 2^{3}-1\} \Big{\}}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \quad\bigcup \limits_{ 1\leq i \leq \lfloor \frac{7q}{8} \rfloor, \; j = 0}^ {\lfloor \frac{7q-8i}{2q} \rfloor}&&\Big{\{}\{ 2^{5} q l + 14q-16i-4qj, \; \; l = 0, 1, 2 , \cdots, 2^{2}-1 \}\cup \{ 2^{7}q-2^{5} q l - \\&&14q+16i+4qj, l = 0, 1, 2 , \cdots, 2^{2}-1\}\bigcup \{ 2^{5} q l + 2^{3}q+2^{2}q+ 14q\\&&-16i-4qj, \; \; l = 0, 1, 2 , \cdots, 2^{2}-1 \}\cup \{ 2^{7}q-2^{5} q l - 2^{3}q-2^{2}q- \\&&14q+16i+4qj, \; \; l = 0, 1, 2 , \cdots, 2^{2}-1\} \Big{\}}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \bigcup \limits_{ l = i}^ {\frac{q-1}{2} }&& \Big{\{}\{ 2^{3} q i + 8l-4, \; \; i = 0, 1, 2 , \cdots, 2^{4}-1 \}\cup \{ 2^{7}q-2^{3} q i - 8l+4, \; \; \\&&i = 0, 1, 2 , \cdots, 2^{4}-1\} \Big{\}}. \end{eqnarray*}$

For $\beta = 8,$ the following congruence equations turning out to be

$\begin{eqnarray} \eta^{2} &\equiv& (qt)^{2}\; (\text{mod} \; 2^{8}q), \; \; t = 2i-1, \; i = 1, 2, 3, \cdots, 2^{5}, \end{eqnarray}$

(3.17)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{8}q), \; \; t = pl, \; \; l = 2i-1 \; i = 1, 2, 3 , \cdots, 2^{3}, \end{eqnarray}$

(3.18)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{8}q), \; \; t = 2qi, \; \; i = 0, 1, 2, , \cdots, 2^{2}-1, \end{eqnarray}$

(3.19)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{8}q), \; \; t = 15q-16i-2qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{15q}{16} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 2, \cdots, \lfloor \frac{15q-16i}{2q} \rfloor, \end{eqnarray}$

(3.20)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{8}q), \; \; t = 8l, \; \; l = 1, 2 \cdots \frac{q-1}{2} , \end{eqnarray}$

(3.21)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{8}q), \; \; t = 6q-8i-2qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{6q}{8} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 2, \cdots, \lfloor \frac{6q-8i}{2q} \rfloor. \end{eqnarray}$

(3.22)

Zeroes of congruences (3.17) $-$ (3.22) are

$\begin{eqnarray*} \bigcup \limits_{t = 2i-1, i = 1}^{\; 2^{5}}&& \{ qt, 2^{7}q + qt \}\bigcup \{2^{8}q-qt, 2^{8}q-2^{7}q-qt\}, \end{eqnarray*}$

$\begin{eqnarray*} \bigcup \limits_{t = q(2i-1), i = 1}^{\; 2^{3}} && \{ 2^{6}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{2}-1 \}\bigcup \{2^{8}q-2^{6}qj-2t, j = 0, \\&& 1, 2, \cdots, 2^{2}-1\}, \end{eqnarray*}$

$\begin{eqnarray*} \bigcup \limits_{k = 2qi, \; i = 1}^{3} && \{ 2^{5}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{3}-1 \}\bigcup \{2^{8}q-2^{5}qj-2t, j = 0, \\&& 1, 2, \cdots, 2^{3}-1\}, \{ 2^{4}qj, \; \; j = 0, 1, 2, \cdots, 2^{4}-1 \}, \end{eqnarray*}$

$\begin{eqnarray*} \bigcup \limits_{ 1\leq i \leq \lfloor 15q/16 \rfloor, \; j = 0}^ {\lfloor (15q-16i)/2q \rfloor} &&\Big{\{}\{ 2^{6} q l + 30q-32i-4qj, \; l = 0, 1, \cdots, 2^{2}-1 \}\cup \{ 2^{8}q-2^{6} q l \\&& - 30q+32i+4qj, l = 0, 1, \cdots, 2^{2}-1\}\bigcup\{ 2^{6} ql +2^{3}q+2^{2}q+ \\&& 30q- 32i-4qj, \; l = 0, 1, \cdots, 2^{2}-1 \}\bigcup \{ 2^{8}q-2^{6} q l -2^{3}q- \\&&2^{2}q- 30q+ 32i+ 4qj, \; l = 0, 1, \cdots, 2^{2}-1\} \Big{\}}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \bigcup \limits_{ 1\leq i \leq \lfloor 15q/16 \rfloor, \; j = 0}^ {\lfloor (15q-16i)/2q \rfloor} &&\Big{\{}\{ 2^{6} q l + 30q-32i-4qj, \; l = 0, 1, \cdots, 2^{2}-1 \}\cup \{ 2^{8}q-2^{6} q l -\\&& 30q+32i+4qj, l = 0, 1, \cdots, 2^{2}-1\}\bigcup\{ 2^{6} q l +2^{3}q+2^{2}q+ 30q \\&&- 32i-4qj, \; l = 0, 1, \cdots, 2^{2}-1 \}\cup \{ 2^{8}q-2^{6} q l -2^{3}q-2^{2}q- \\&&30q+ 32i+ 4qj, \; l = 0, 1, \cdots, 2^{2}-1\} \Big{\}}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \bigcup \limits_{t = 8l, \; l = 1}^{(q-1)/2}&& \{ 2^{4}qj+2t, \; j = 0, 1, 2, \cdots, 2^{4}-1 \}\bigcup \{ 2^{8}q-2^{4}qj-2t, j = 0, 1, \\&& 2, \cdots, 2^{4}-1\}, \\ \end{eqnarray*}$

$\begin{eqnarray*} \quad \; \bigcup \limits_{ 1\leq i \leq \lfloor 6q/8 \rfloor, \; j = 0}^ {\lfloor (6q-8i)/2q \rfloor}&& \Big{\{}\{ 2^{4} q l + 12q-16i-4qj, l = 0, 1, 2 , \cdots, 2^{4}-1 \}\cup \{ 2^{8}q- 2^{4} q l - \\ &&12q+16i+ 4qj, l = 0, 1, 2 , \cdots, 2^{4}-1\} \Big{\}}. \end{eqnarray*}$

For $\beta = 9,$ we have

$\begin{eqnarray} \eta^{2} &\equiv& (qt)^{2}\; (\text{mod} \; 2^{9}q), \; \; t = 2i-1, \; i = 1, 2, 3, \cdots, 2^{6}, \end{eqnarray}$

(3.23)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{9}q), \; \; t = ql, \; \; l = 2i-1 \; i = 1, 2, 3 , \cdots, 2^{4}, \end{eqnarray}$

(3.24)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{9}q), \; \; t = 2qi, \; i = 0, 1, 3, 4, 5, 7, \end{eqnarray}$

(3.25)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{9}q), \; \; t = 31q-32i-2qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{31p}{32} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 3, \cdots, \lfloor \frac{31q-32i}{2q} \rfloor, \end{eqnarray}$

(3.26)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{9}q), \; \; t = 4q, \end{eqnarray}$

(3.27)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{9}q), \; \; t = 8l, \; \; l = 1, 2, \cdots, q-1, \end{eqnarray}$

(3.28)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{9}q), \; \; t = 14q-16i-4qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{14q}{16} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 3, \cdots, \lfloor \frac{14q-16i}{4q} \rfloor, \end{eqnarray}$

(3.29)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{9}q), \; \; t = 8l-4, \; \; l = 1, 2 \cdots \frac{q-1}{2}. \end{eqnarray}$

(3.30)

Sets of solution of congruences (3.23) $-$ (3.30) are

$\begin{eqnarray*} \bigcup \limits_{t = 2i-1, i = 1}^{\; 2^{6}}&& \{ qt, 2^{8}q+qt \}\bigcup \{2^{9}q-qt, 2^{9}q-2^{8}q-qt\}, \end{eqnarray*}$

$\begin{eqnarray*} \bigcup \limits_{t = q(2i-1), \; i = 1}^{\; 2^{4}} &&\{ 2^{7}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{2}-1 \}\bigcup \{2^{9}q-2^{7}qj-2t, j = 0, \\ && 1, 2, \cdots, 2^{2}-1\}, \end{eqnarray*}$

$\begin{eqnarray*} \bigcup\limits_{\substack{t = 2qi, \\i \in \{ 1, 3, 4, 5, 7\}}} && \{ 2^{6}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{3}-1 \}\bigcup \{2^{9}q-2^{6}qj-2t, j = 0, \\ && 1, 2, \cdots, 2^{3}-1\}, \{ 2^{5}qj, \; \; j = 0, 1, 2, \cdots, 2^{4}-1 \}, \end{eqnarray*}$

$\begin{eqnarray} \quad \quad \bigcup \limits_{ 1\leq i \leq \lfloor 31p/32 \rfloor, \; j = 0}^ {\lfloor (31q-32i)/2q \rfloor}&& \Big{\{}\{ 2^{7} q l + 62q-64i-4qj, \; \; l = 0, 1, 2 , \cdots, 2^{2}-1 \}\cup \{ 2^{9}q- \end{eqnarray}$

(3.31)

$\begin{eqnarray} && 2^{7} q l - 62q+64i+4qj, l = 0, 1, 2 , \cdots, 2^{2}-1\}\bigcup\{ 2^{7} q l+2^{6}q + \end{eqnarray}$

(3.32)

$\begin{eqnarray} &&2^{3}q+2^{2}q+ 62q-64i-4qj, \; \; l = 0, 1, 2 , \cdots, 2^{2}-1 \}\cup \{ 2^{9}q- \end{eqnarray}$

(3.33)

$\begin{eqnarray} && 2^{7} q l - 2^{6}q -2^{3}q-2^{2}q- 62q+64i+4qj, \; \; l = 0, 1, 2 , \cdots, 2^{2}-1\} \Big{\}}, \end{eqnarray}$

(3.34)

$\begin{eqnarray} \bigcup\limits_{\substack{t = 8l, \\1 \leq l \leq q-1}}&& \Big{\{}\{ 2^{5} qj + 2t, \; \; j = 0, 1, 2 , \cdots, 2^{4}-1 \}\cup \{ 2^{9}q- 2^{5} q j - 2t, j = 0, \end{eqnarray}$

(3.35)

$\begin{eqnarray} &&1, 2 , \cdots, 2^{4}-1\} \Big{\}}, \{2^{4}qj+2t, t = 4q, \; j = 0, 1, 2, \cdots, 2^{5}-1 \}, \end{eqnarray}$

(3.36)

$\begin{eqnarray*} \quad \bigcup \limits_{ 1\leq i \leq \lfloor 14q/16 \rfloor, \; j = 0}^ {\lfloor (14q-16i)/4q \rfloor} && \Big{\{}\{ 2^{6} q l + 28q-32i-8qj, \; l = 0, 1, 2 , \cdots, 2^{3}-1 \}\cup \{ 2^{9}q-2^{6} q l \\ &&- 28q+32i+8qj, l = 0, 1, 2 , \cdots, 2^{3}-1\}\bigcup\{ 2^{6} q l +2^{4}q+2^{3}q + \\ &&28q-32i- 8qj, \; l = 0, 1, 2 , \cdots, 2^{3}-1 \}\cup \{ 2^{9}q-2^{6} q l -2^{4}q-\\ &&2^{3}q- 28q+32i+8qj, l = 0, 1, 2 , \cdots, 2^{3}-1\} \Big{\}}, \end{eqnarray*}$

$\begin{eqnarray*} \; \bigcup \limits_{ l = 1}^{(q-1)/2 }&&\Big{\{}\{ 2^{4} q i + 16l-8, \; \; i = 0, 1, 2 , \cdots, 2^{5}-1 \}\cup \{ 2^{9}q-2^{4} q i - 16l+\\ &&8, i = 0, 1, 2 , \cdots, 2^{5}-1\} \Big{\}}. \end{eqnarray*}$

For $\beta = 10,$ we get

$\begin{eqnarray} \eta^{2} &\equiv& (qt)^{2}\; (\text{mod} \; 2^{10}q), \; \; t = 2i-1, \; i = 1, 2, 3, \cdots, 2^{7}, \end{eqnarray}$

(3.37)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{10}q), \; \; t = ql, \; \; l = 2i-1 \; i = 1, 2, 3 , \cdots, 2^{5}, \end{eqnarray}$

(3.38)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{10}q), \; \; t = 2qi, \; i = 2j-1, \; \; j = 1, 3, \cdots, 2^{3} , \end{eqnarray}$

(3.39)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{10}q), \; \; t = 63q-64i-2qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{63q}{64} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 3, \cdots, \lfloor \frac{63q-64i}{2q} \rfloor, \end{eqnarray}$

(3.40)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{10}q), \; \; t = 4qi, \; \; i = 0, 1, 2, , \cdots, 2^{2}-1, \end{eqnarray}$

(3.41)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{10}q), \; \; t = 30q-32i-4qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{30q}{32} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 3, \cdots, \lfloor \frac{30q-32i}{4q} \rfloor, \end{eqnarray}$

(3.42)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{10}q), \; \; t = 16l, \; \; l = 1, 2 \cdots \frac{q-1}{2} , \end{eqnarray}$

(3.43)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{10}q), \; \; t = 12q-16i-4qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{12q}{16} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 3, \cdots, \lfloor \frac{12q-16i}{4q} \rfloor. \end{eqnarray}$

(3.44)

Zeroes of congruences (3.37) $-$ (3.44) are

$\begin{eqnarray*} \bigcup \limits_{t = 2i-1, \; i = 1}^{\; 2^{7}} &&\{ qt, 2^{9}+qt \}\bigcup \{2^{10}q-qt, 2^{10}q-2^{9}q-qt\}, \end{eqnarray*}$

$\begin{eqnarray*} \quad\quad\quad\; \bigcup \limits_{t = q(2i-1), i = 1}^{\; 2^{5}}&& \{ 2^{8}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{2}-1 \}\bigcup \{2^{10}q-2^{8}qj-2t, j = 0, 1, 2, \\&& \cdots, 2^{2}-1\}, \end{eqnarray*}$

$\begin{eqnarray*} \quad\quad\quad\; \bigcup \limits_{t = 2q(2i-1), i = 1}^{\; 2^{3}}&& \{ 2^{7}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{3}-1 \}\bigcup \{2^{10}q-2^{7}qj-2t, j = 0, 1, 2, \\&& \cdots, 2^{3}-1\}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \bigcup \limits_{t = 4qi, \; i = 1}^{3} &&\{ 2^{6}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{4}-1 \}\bigcup \{2^{10}q-2^{6}qj-2t, j = 0, \\&& 1, 2, \cdots, 2^{4}-1\}, \{ 2^{5}qj, \; \; j = 0, 1, 2, \cdots, 2^{5}-1 \}, \end{eqnarray*}$

$\begin{eqnarray*} \quad\; \; \bigcup \limits_{ 1\leq i \leq \lfloor 30q/32 \rfloor, \; j = 0}^ {\lfloor (30q-32i)/4q \rfloor}&& \Big{\{}\{ 2^{7} q t + 60q- 64i-8qj, \; \; l = 0, 1, 2 , \cdots, 2^{3}-1 \}\cup \{ 2^{10}q- \\&&2^{7} q l - 60q+64i+8qj, l = 0, 1, 2 , \cdots, 2^{3}-1\}\bigcup \{ 2^{7} q l + 2^{4}q+ \\&&2^{3}q+ 60q- 64i-8qj, \; \; l = 0, 1, 2 , \cdots, 2^{3}-1 \}\cup \{ 2^{10}q- 2^{7} q l -\\&&2^{4}q- 2^{3}q - 60q+64i+8qj, l = 0, 1, 2 , \cdots, 2^{3}-1\} \Big{\}}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \bigcup \limits_{t = 16l, l = 1}^{(q-1)/2} && \{ 2^{5}qj+2t, \; j = 0, 1, 2, \cdots, 2^{5}-1\}\bigcup \{ 2^{10}q-2^{5}qj-2t, j = 0, \\&& 1, 2, \cdots, 2^{5}-1\}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \bigcup \limits_{ 1\leq i \leq \lfloor 12q/16 \rfloor, \; j = 0}^ {\lfloor (12q-16i)/4q \rfloor} && \Big{\{}\{ 2^{5} q l + 24q-32i-8qj, l = 0, 1, 2 , \cdots, 2^{5}-1 \}\cup \{ 2^{10}q-2^{5} q l \\&&- 24q+32i+8qj, l = 0, 1, 2 , \cdots, 2^{5}-1\} \Big{\}}.\nonumber \end{eqnarray*}$

For $\beta = 11,$ we obtain

$\begin{eqnarray} \eta^{2} &\equiv& (qt)^{2}\; (\text{mod} \; 2^{11}q), \; \; t = 2i-1, \; i = 1, 2, 3, \cdots, 2^{8}, \end{eqnarray}$

(3.45)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{11}q), \; \; t = ql, \; \; l = 2i-1 \; i = 1, 2, 3 , \cdots, 2^{6}, \end{eqnarray}$

(3.46)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{11}q), \; \; t = 2qi, \; i = 2j-1, \; \; j = 1, 3, \cdots, 2^{4} , \end{eqnarray}$

(3.47)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{11}q), \; \; t = 127q-128i-2qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{127q}{128} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 3, \cdots, \lfloor \frac{127q-128i}{2q} \rfloor, \end{eqnarray}$

(3.48)

$\begin{eqnarray} \quad\; \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{11}q), \; \; t = 4qi, \; \; i = 0, 1, 3, 4, 5, 7, \end{eqnarray}$

(3.49)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{11}q), \; \; t = 62q-64i-4qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{62q}{64} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 3, \cdots, \lfloor \frac{62q-64i}{4q} \rfloor, \end{eqnarray}$

(3.50)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{11}q), \; \; t = 8q, \end{eqnarray}$

(3.51)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{11}q), \; \; t = 16l, \; \; l = 1, 2, \cdots, q-1, \end{eqnarray}$

(3.52)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{11}q), \; \; t = 28q-32i-8qj, \; i = 1, 2, 3, \cdots, \lfloor \frac{28q}{32} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 3, \cdots, \lfloor \frac{28q-32i}{8q} \rfloor, \end{eqnarray}$

(3.53)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{11}q), \; \; t = 16l-8, \; \; l = 1, 2 \cdots \frac{q-1}{2}. \end{eqnarray}$

(3.54)

Sets of solution of congruences (3.45) $-$ (3.54) are

$\begin{eqnarray*} \bigcup \limits_{t = 2q(2i-1), i = 1}^{\; 2^{4}}&& \{ 2^{8}qj+2t, \; \; j = 0, 1, \\&& 2, \cdots, 2^{3}-1 \}\bigcup \{2^{11}q-2^{8}qj-2t, j = 0, 1, 2, \cdots, 2^{3}-1\}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \bigcup \limits_{ 1\leq i \leq \lfloor 127q/128 \rfloor, \; j = 0}^ {\lfloor (127q-128i)/2q \rfloor}&&\Big{\{}\{ 2^{9} qt + 254q-256i-4qj, \; \; t = 0, 1, \cdots, 2^{2}-1 \}\cup \{ 2^{11}q- \\&&2^{9} q l - 254q+256i+4qj, l = 0, 1, \cdots, 2^{2}-1\} \bigcup \{ 2^{9} q l+2^{7}q\\&&+ 2^{6}q+2^{3}q+2^{2}q + 254q- 256i-4qj, \; \; l = 0, 1, \cdots, 2^{2}-1 \}\\&& \cup \{ 2^{11}q- 2^{9} q l - 2^{7}q-2^{6}q-2^{3}q-2^{2}q - 254q+256i+4qj, \\&& l = 0, 1, \cdots, 2^{2}-1\} \Big{\}}, \end{eqnarray*}$

$\begin{eqnarray*} \; \bigcup\limits_{\substack{t = 2qi, \\ i \in \{ 1, 3, 4, 5, 7\}}} && \{ 2^{7}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{4}-1 \}\bigcup \{2^{11}q-2^{7}qj-2t, j = \\&& 0, 1, 2, \cdots, 2^{4}-1\}, \{ 2^{6}qj, \; \; j = 0, 1, 2, \cdots, 2^{5}-1 \}, \end{eqnarray*}$

$\begin{eqnarray*} \quad\bigcup \limits_{ 1\leq i \leq \lfloor 62q/64 \rfloor, \; j = 0}^ {\lfloor (62q-64i)/4q \rfloor}&& \Big{\{}\{ 2^{8} q l + 124q-128i-8qj, \; \; l = 0, 1, 2 , \cdots, 2^{3}-1 \}\cup \{ 2^{11}q\\&&- 2^{8} q l - 124q+128i+8qj, l = 0, 1, 2 , \cdots, 2^{3}-1\}\bigcup\{ 2^{8} q l+\\&&2^{7}q +2^{4}q+2^{3}q+ 124q-128i-8qj, \; \; l = 0, 1, 2 , \cdots, 2^{3}-1 \}\\ &&\cup \{ 2^{11}q- 2^{8} q l - 2^{7}q -2^{4}q-2^{3}q- 124q+128i+8qj, \; \; l = 0, \\ && 1, 2 , \cdots, 2^{3}-1\} \Big{\}}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \bigcup\limits_{\substack{t = 16l, \\1 \leq l \leq q-1}}&& \Big{\{}\{ 2^{6} q j + 2t, \; \; j = 0, 1, 2 , \cdots, 2^{5}-1 \}\cup \{ 2^{11}q- 2^{6} q j - 2t, j = \\ &&0, 1, 2 , \cdots, 2^{5}-1\} \Big{\}}, \{2^{5}qj+2t, t = 8q, \; j = 0, 1, 2, \cdots, 2^{6}-1 \}, \end{eqnarray*}$

$\begin{eqnarray*} \quad\quad \bigcup \limits_{ 1\leq i \leq \lfloor 28q/32 \rfloor, \; j = 0}^ {\lfloor (14q-16i)/8q \rfloor} && \Big{\{}\{ 2^{7} q l + 56q-64i-16qj, \; l = 0, 1, 2 , \cdots, 2^{4}-1 \}\cup \{ 2^{11}q-2^{7} q l \\ &&- 56q+64i+16qj, l = 0, 1, 2 , \cdots, 2^{4}-1\}\bigcup\{ 2^{7} q l +2^{5}q+2^{4}q + \\ &&56q-64i- 16qj, \; l = 0, 1, 2 , \cdots, 2^{4}-1 \}\cup \{ 2^{11}q-2^{7} q l -2^{5}q-\\ &&2^{4}q- 56q+64i+16qj, l = 0, 1, 2 , \cdots, 2^{4}-1\} \Big{\}}, \end{eqnarray*}$

$\begin{eqnarray*} \quad\quad \bigcup \limits_{ l = 1}^{(q-1)/2 }&&\Big{\{}\{ 2^{5} q i + 32l-16, \; \; i = 0, 1, 2 , \cdots, 2^{6}-1 \}\cup \{ 2^{11}q-2^{5} q i - 32l+\\ &&16, i = 0, 1, 2 , \cdots, 2^{6}-1\} \Big{\}}. \end{eqnarray*}$

For $\beta\geq10$ with $\beta\equiv 0\; (\text{mod}\; 2)$ , there are $2^{\beta-5}5+\frac{(2^{\beta-8}-1)2^{\beta-5}}{3\cdot2^{\beta-8}}(2q-1)+6q-2$ congruences. We have

$\begin{eqnarray} \eta^{2} &\equiv& (pt)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = 2i-1, \; i = 1, 2, 3, \cdots, 2^{\beta-3}, \end{eqnarray}$

(3.55)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = ql, \; \; l = 2i-1 \; i = 1, 2, 3 , \cdots, 2^{\beta-5}. \end{eqnarray}$

(3.56)

For $\nu = 1, 2, 3, \cdots, \frac{\beta-8}{2},$

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = 2^{\nu}qi, \; i = 2j-1, \; \; j = 1, 3, \cdots, 2^{\beta-7}, \end{eqnarray}$

(3.57)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = \Big{(}2^{\beta-5+\nu}~~-2^{\nu-1}~~\Big{)}q- (2^{\beta-5+\nu}~~)i-2^{\nu}qj, \\ && \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; i = 1, 2, \cdots, \lfloor \frac{2^{\beta-5+\nu}~~-2^{\nu-1}~~q}{2^{\beta-5+\nu}~~} \rfloor, \\ && \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, \cdots, \lfloor \frac{(2^{\beta-5+\nu}~~-2^{\nu-1}~~)q-(2^{\beta-5+\nu}~~)i}{2^{\nu}q} \rfloor, \end{eqnarray}$

(3.58)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = 2^{\frac{\beta-6}{2}}qi, \; \; i = 0, 1, 2, , \cdots, 2^{2}-1, \end{eqnarray}$

(3.59)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = (2^{\frac{\beta}{2}}-2^{\frac{\beta-8}{2}})q-2^{\frac{\beta}{2}}i-2^{\frac{\beta-6}{2}}qj, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; i = 1, 2, 3, \cdots, \lfloor \frac{(2^{\frac{\beta}{2}}-2^{\frac{\beta-8}{2}})q}{2^{\frac{\beta}{2}}} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 2, \cdots, \lfloor \frac{(2^{\frac{\beta}{2}}-2^{\frac{\beta-8}{2}})q-2^{\frac{\beta}{2}}i}{2^{\frac{\beta-6}{2}}q} \rfloor, \end{eqnarray}$

(3.60)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = 2^{\frac{\beta-2}{2}}l, \; \; l = 1, 2 \cdots \frac{q-1}{2} , \end{eqnarray}$

(3.61)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = (2^{\frac{\beta-2}{2}}-2^{\frac{\beta-6}{2}})q-2^{\frac{\beta-2}{2}}i-2^{\frac{\beta-6}{2}}qj, \\ && \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; i = 1, 2, 3, \cdots, \lfloor \frac{(2^{\frac{\beta-2}{2}}-2^{\frac{\beta-6}{2}})q}{2^{\frac{\beta-2}{2}}} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 2, \cdots, \lfloor \frac{(2^{\frac{\beta-2}{2}}-2^{\frac{\beta-6}{2}})q-2^{\frac{\beta-2}{2}}i}{2^{\frac{\beta-6}{2}}q} \rfloor. \end{eqnarray}$

(3.62)

Zeroes of congruences (3.55) $-$ (3.62) are

$\begin{eqnarray*} \quad \quad \quad\; \bigcup \limits_{t = 2i-1, \; i = 1}^{\; 2^{\beta-3}}&& \{ qt, 2^{\beta-1}~~q+qt \}\bigcup \{2^{\beta}q-qt, 2^{\beta-1}~~q-2^{\beta-1}~~q-qt\}, \\ \bigcup \limits_{t = q(2i-1), i = 1}^{\; 2^{\beta-5}}&& \{ 2^{\beta-2}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{2}-1 \}\bigcup \{2^{\beta}q-2^{\beta-2}qj-2t, j = 0, \\&& 1, 2, \cdots, 2^{2}-1\}, \end{eqnarray*}$

For $\nu = 1, 2, 3, \cdots, \frac{\beta-8}{2},$

$\begin{eqnarray*} \bigcup \limits_{t = 2^{\nu}q(2i-1), i = 1}^{2^{\beta-7}} \bigcup \limits_{\nu = 1}^{(\beta-8)/2}&& \{ 2^{\beta-2-\nu}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{3}-1 \}\bigcup \{2^{\beta}q-2^{\beta-2-\nu}qj\\&&-2t, j = 0, 1, 2, \cdots, 2^{3}-1\}, \end{eqnarray*}$

$\begin{eqnarray*} \; \bigcup \limits_{ 1\leq i \leq \lfloor A \rfloor, \; j = 0}^ {\lfloor B \rfloor}\bigcup \limits_{\nu = 1}^{(\beta-8)/2}&&\Big{\{}\{ 2^{\beta-1-\nu} q l + 2t, \; \; l = 0, 1, \cdots, 2^{2}-1 \}\cup \{ 2^{\beta}q- 2^{\beta-1-\nu} q l \\&&- 2t, l = 0, 1, \cdots, 2^{2}-1\} \bigcup \{ 2^{\beta-1-\nu} q k+2^{\nu-1}~~C+ 2t, l = 0, \\&& 1, \cdots, 2^{2}-1 \}\cup \{ 2^{\beta}q- 2^{\beta-1-\nu} q l - 2^{\nu-1}~~C-2t, l = 0, 1, \cdots, \\&&2^{2}-1\} \Big{\}}, \end{eqnarray*}$

where $A = \lfloor \frac{2^{\beta-5+\nu}~~-2^{\nu-1}~~q}{2^{\beta-5+\nu}~~} \rfloor$ , $B = \lfloor \frac{(2^{\beta-5+\nu}~~-2^{\nu-1}~~)q-(2^{\beta-5+\nu}~~)i}{2^{\nu}q} \rfloor$ , $C = 2^{7}q+2^{6}q+2^{3}q+2^{2}q$ and $t = \Big{(}2^{\beta-5+\nu}~~-2^{\nu-1}~~\Big{)}q- (2^{\beta-5+\nu}~~)i-2^{\nu}qj$ .

$\begin{eqnarray*} \; \; \; \bigcup \limits_{t = 2^{\frac{\beta-6}{2}}qi, \; i = 1}^{3} &&\{ 2^{\frac{\beta+2}{2}}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{\frac{\beta-2}{2}}-1 \} \bigcup \{2^{\beta}q-2^{\frac{\beta+2}{2}}qj-2t, j = \\&& 0, 1, 2, \cdots, 2^{\frac{\beta-2}{2}}-1\}, \{ 2^{\frac{\beta}{2}}qj, \; \; j = 0, 1, 2, \cdots, 2^{\frac{\beta}{2}}-1 \}, \\ \bigcup \limits_{ 1\leq i \leq \lfloor D \rfloor, \; j = 0}^ {\lfloor E \rfloor}&& \Big{\{}\{ 2^{\frac{\beta+4}{2}} q l + 2t, \; \; l = 0, 1, 2 , \cdots, 2^{\frac{\beta-4}{2}}-1 \}\cup \{ 2^{\beta}q- 2^{\frac{\beta+4}{2}} q l - 2t, \; \; l = \\&&0, 1, 2 , \cdots, 2^{\frac{\beta-4}{2}}-1\}\bigcup \{ 2^{\frac{\beta+4}{2}} q l +2^{(\beta-10)/2}F+ 2t, \; l = 0, 1, 2 , \cdots, \\&& 2^{\frac{\beta-4}{2}}-1 \}\cup \{ 2^{\beta}q- -2^{(\beta-10)/2}F- 2t, l = 0, 1, 2 , \cdots, 2^{\frac{\beta-4}{2}}-1\} \Big{\}}, \nonumber \end{eqnarray*}$

where $D = \frac{(2^{\frac{\beta}{2}}-2^{\frac{\beta-8}{2}})q}{2^{\frac{\beta}{2}}}$ , $E = \frac{(2^{\frac{\beta}{2}}-2^{\frac{\beta-8}{2}})q-2^{\frac{\beta}{2}}i}{2^{\frac{\beta-6}{2}}q}$ , $F = 2^{4}q+2^{3}q$ and $t = (2^{\frac{\beta}{2}}-2^{\frac{\beta-8}{2}})q-2^{\frac{\beta}{2}}i-2^{\frac{\beta-6}{2}}qj$ .

$\begin{eqnarray*} \; \; \bigcup \limits_{t = 2^{(\beta-2)/2}l, l = 1}^{(q-1)/2} && \{ 2^{\beta/2}qj+2t, \; j = 0, 1, 2, \cdots, 2^{\beta/2}-1\}\bigcup \{ 2^{\beta}q-2^{\beta/2}qj-2t, j = \\&&0, 1, 2, \cdots, 2^{\beta/2}-1\}, \\ \bigcup \limits_{ 1\leq i \leq \lfloor G \rfloor, \; j = 0}^ {\lfloor H \rfloor} && \Big{\{}\{ 2^{\beta/2} q l + 2t, \; \; l = 0, 1, 2 , \cdots, 2^{\beta/2}-1 \}\cup \{ 2^{\beta}q-2^{\beta} q l - 2t, \; \; l = \\&&0, 1, 2 , \cdots, 2^{\beta/2}-1\} \Big{\}}, \end{eqnarray*}$

where $G = \frac{(2^{\frac{\beta-2}{2}}-2^{\frac{\beta-6}{2}})q}{2^{\frac{\beta-2}{2}}}$ , $H = \frac{(2^{\frac{\beta-2}{2}}-2^{\frac{\beta-6}{2}})q-2^{\frac{\beta-2}{2}}i}{2^{\frac{\beta-6}{2}}q}$ and $t = (2^{\frac{\beta-2}{2}}-2^{\frac{\beta-6}{2}})q-2^{\frac{\beta-2}{2}}i-2^{\frac{\beta-6}{2}}qj$ .

For $\beta\geq11$ with $\beta\equiv 1\; (\text{mod}\; 2),$ there are $2^{\beta-5}5+\frac{(2^{\beta-8}-1)2^{\beta-5}}{3\cdot2^{\beta-8}}(2q-1)+ \frac{23q-9}{2}$ congruences. We have

$\begin{eqnarray} \eta^{2} &\equiv& (qt)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = 2i-1, \; i = 1, 2, 3, \cdots, 2^{\beta-3}, \end{eqnarray}$

(3.63)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = ql, \; \; l = 2i-1 \; i = 1, 2, 3 , \cdots, 2^{\beta-5}. \end{eqnarray}$

(3.64)

For $\nu = 1, 2, 3, \cdots, \frac{\beta-9}{2},$

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = 2^{\nu}qi, \; i = 2j-1, \; \; j = 1, 3, \cdots, 2^{\beta-7}, \end{eqnarray}$

(3.65)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = \Big{(}2^{\beta-5+\nu}~~-2^{\nu-1}~~\Big{)}q- (2^{\beta-5+\nu}~~)i-2^{\nu}qj, \\ && \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; i = 1, 2, \cdots, \lfloor \frac{2^{\beta-5+\nu}~~-2^{\nu-1}~~q}{2^{\beta-5+\nu}~~} \rfloor, \\ && \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, \cdots, \lfloor \frac{(2^{\beta-5+\nu}~~-2^{\nu-1}~~)q-(2^{\beta-5+\nu}~~)i}{2^{\nu}q} \rfloor, \end{eqnarray}$

(3.66)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = 2^{\frac{\beta+1}{2}}qi, \; \; i = 0, 1, 3, 4, 5, 7, \end{eqnarray}$

(3.67)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = (2^{\frac{\beta+1}{2}}-2^{{\frac{\beta-9}{2}}})q-2^{\frac{\beta+1}{2}}i-2^{\frac{\beta-7}{2}}qj, \\ && \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; i = 1, 2, 3, \cdots, \lfloor \frac{2^{\frac{\beta+1}{2}}-2^{{\frac{\beta-9}{2}}})q}{2^{\frac{\beta+1}{2}}} \rfloor, \\ && \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 3, \cdots, \lfloor \frac{(2^{\frac{\beta+1}{2}}-2^{{\frac{\beta-9}{2}}})q-2^{\frac{\beta+1}{2}}i}{2^{\frac{\beta-7}{2}}q} \rfloor, \end{eqnarray}$

(3.68)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = 2^{\frac{\beta-5}{2}}q, \end{eqnarray}$

(3.69)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = 2^{\frac{\beta-3}{2}}l, \; \; l = 1, 2, \cdots, q-1, \end{eqnarray}$

(3.70)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = (2^{\frac{\beta-1}{2}}-2^{\frac{\beta-7}{2}})q-2^{\frac{\beta-1}{2}}i-2^{\frac{\beta-5}{2}}qj, \\ && \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; i = 1, 2, 3, \cdots, \lfloor \frac{(2^{\frac{\beta-1}{2}}-2^{\frac{\beta-7}{2}})q}{2^{\frac{\beta-1}{2}}} \rfloor, \\ &&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; j = 0, 1, 3, \cdots, \lfloor \frac{(2^{\frac{\beta-1}{2}}-2^{\frac{\beta-7}{2}})q-2^{\frac{\beta-1}{2}}i}{2^{\frac{\beta-5}{2}}q} \rfloor, \end{eqnarray}$

(3.71)

$\begin{eqnarray} \eta^{2} &\equiv& (2t)^{2}\; (\text{mod} \; 2^{\beta}q), \; \; t = 2^{\frac{\beta-3}{2}}l-2^{\frac{\beta-5}{2}}, \; \; l = 1, 2 \cdots \frac{q-1}{2}. \end{eqnarray}$

(3.72)

Zeroes of congruences (3.63) $-$ (3.72) are

$\begin{eqnarray*} \bigcup \limits_{t = 2i-1, \; i = 1}^{\; 2^{\beta-3}} &&\{ qt, 2^{\beta-1}~~q+qt \}\bigcup \{2^{\beta}q-qt, 2^{\beta-1}~~q-2^{\beta-1}~~q-qt\}, \\ \bigcup \limits_{t = q(2i-1), i = 1}^{\; 2^{\beta-5}}&& \{ 2^{\beta-2}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{2}-1 \}\bigcup \{2^{\beta}q-2^{\beta-2}qj\\&&-2t, j = 0, 1, 2, \cdots, 2^{2}-1\}, \end{eqnarray*}$

For $\nu = 1, 2, 3, \cdots, \frac{\beta-9}{2},$

$\begin{eqnarray*} \bigcup \limits_{t = 2^{\nu}q(2i-1), i = 1}^{2^{\beta-7}} \bigcup \limits_{\nu = 1}^{(\beta-9)/2}&& \{ 2^{\beta-2-\nu}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{3}-1 \}\bigcup \{2^{\beta}q-\\&&2^{\beta-2-\nu}qj-2t, j = 0, 1, 2, \cdots, 2^{3}-1\}, \\ \bigcup \limits_{ 1\leq i \leq \lfloor A \rfloor, \; j = 0}^ {\lfloor B \rfloor}\bigcup \limits_{\nu = 1}^{(\beta-9)/2}&&\Big{\{}\{ 2^{\beta-1-\nu} q l + 2t, \; \; l = 0, 1, \cdots, 2^{2}-1 \}\cup \{ 2^{\beta}q- \\&&2^{\beta-1-\nu} q l - 2t, \; \; l = 0, 1, \cdots, 2^{2}-1\} \bigcup \{ 2^{\beta-1-\nu} q t+\\&&2^{\nu-1}~~C+ 2t, l = 0, 1, \cdots, 2^{2}-1 \}\cup \{ 2^{\beta}q- 2^{\beta-1-\nu} q l \\&&- 2^{\nu-1}~~C-2t, l = 0, 1, \cdots, 2^{2}-1\} \Big{\}}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \bigcup\limits_{\substack{t = 2^{(\beta+1)/2}qi, \\ i \in \{ 1, 3, 4, 5, 7\}}} && \{ 2^{(\beta+3)/2}qj+2t, \; \; j = 0, 1, 2, \cdots, 2^{(\beta-3)/2}-1 \}\bigcup \{2^{\beta}q-2^{(\beta+3)/2}qj\\&&-2t, j = 0, 1, 2, \cdots, 2^{(\beta-3)/2}-1\}, \{ 2^{(\beta+1)/2}qj, \; \; j = 0, 1, 2, \cdots, \\&& 2^{(\beta-1)/2}-1 \}, \end{eqnarray*}$

$\begin{eqnarray*} \quad \quad \bigcup \limits_{ 1\leq i \leq \lfloor L \rfloor, \; j = 0}^ {\lfloor M \rfloor}&& \Big{\{}\{ 2^{(\beta+5)/2} q l + 2t, \; \; l = 0, 1, 2 , \cdots, 2^{(\beta-5)/2}-1 \}\cup \{ 2^{\beta}q- 2^{(\beta+5)/2} q l \\&&- 2t, \; \; l = 0, 1, 2 , \cdots, 2^{(\beta-5)/2}-1\}\bigcup \{ 2^{(\beta+5)/2} q l +2^{(\beta-11)/2}N+ 2t, l\\&& = 0, 1, 2 , \cdots, 2^{(\beta-5)/2}-1 \}\cup \{ 2^{\beta}q- 2^{(\beta+5)/2} q l -2^{(\beta-11)/2}N- 2t, l = \\&&0, 1, 2 , \cdots, 2^{(\beta-5)/2}-1\} \Big{\}}, \end{eqnarray*}$

where $L = \frac{2^{\frac{\beta+1}{2}}-2^{{\frac{\beta-9}{2}}})q}{2^{\frac{\beta+1}{2}}}$ , $M = \frac{(2^{\frac{\beta+1}{2}}-2^{{\frac{\beta-9}{2}}})q-2^{\frac{\beta+1}{2}}i}{2^{\frac{\beta-7}{2}}q}$ , $N = 2^{7}q+2^{4}q+2^{3}q$ and $t = (2^{\frac{\beta+1}{2}}-2^{{\frac{\beta-9}{2}}})q-2^{\frac{\beta+1}{2}}i-2^{\frac{\beta-7}{2}}qj$ .

$\begin{eqnarray*} \bigcup\limits_{\substack{t = 2^{(\beta-3)/2}l, \\1 \leq l \leq q-1}}&& \Big{\{}\{ 2^{(\beta+1)/2} q j + 2t, \; \; j = 0, 1, \cdots, 2^{(\beta-1)/2}-1 \}\cup \{ 2^{\beta}q - 2^{(\beta+1)/2} q j \\&&- 2t, j = 0, 1, \cdots, 2^{(\beta-1)/2}-1\} \Big{\}}, \{2^{(\beta-1)/2} q j+2t, t = (\beta-5)/2q, j\\&& = 0, 1, 2, \cdots, 2^{(\beta+1)/2}-1 \}, \end{eqnarray*}$

$\begin{eqnarray*} \bigcup \limits_{ 1\leq i \leq \lfloor Q \rfloor, \; j = 0}^ {\lfloor R \rfloor} && \Big{\{}\{ 2^{(\beta+3)/2} q l + 2t, \; l = 0, 1, 2 , \cdots, 2^{(\beta-3)/2}-1 \}\cup \{ 2^{\beta}q-2^{(\beta+3)/2} q l \\&&- 2t, l = 0, 1, 2 , \cdots, 2^{(\beta-3)/2}-1\}\bigcup \{ 2^{(\beta+3)/2} q l + 2^{(\beta-11)/2}S+ 2t, l\\&& = 0, 1, 2 , \cdots, 2^{(\beta-3)/2}-1 \}\cup \{ 2^{\beta}q-2^{(\beta+3)/2} q l - 2^{(\beta-11)/2}S- 2t, l\\ && = 0, 1, 2 , \cdots, 2^{(\beta-3)/2}-1\} \Big{\}}, \end{eqnarray*}$

where $Q = \frac{(2^{\frac{\beta-1}{2}}-2^{\frac{\beta-7}{2}})q}{2^{\frac{\beta-1}{2}}}$ , $R = \frac{(2^{\frac{\beta-1}{2}}-2^{\frac{\beta-7}{2}})q-2^{\frac{\beta-1}{2}}i}{2^{\frac{\beta-5}{2}}q}$ , $S = 2^{5}q+2^{4}q$ and $t = (2^{\frac{\beta-1}{2}}-2^{\frac{\beta-7}{2}})q-2^{\frac{\beta-1}{2}}i-2^{\frac{\beta-5}{2}}qj$ .

$\begin{eqnarray*} \bigcup \limits_{ l = 1}^{(q-1)/2 }&&\Big{\{}\{ 2^{(\beta-1)/2} q i + 2^{(\beta-1)/2}l-2^{(\beta-3)/2}, \; \; i = 0, 1, 2 , \cdots, 2^{(\beta+1)/2}-1 \}\cup \{ 2^{\beta}q\\ &&-2^{(\beta-1)/2} q i - 2^{(\beta-1)/2}l+2^{(\beta-3)/2}, i = 0, 1, 2 , \cdots, 2^{(\beta+1)/2}-1\} \Big{\}}. \end{eqnarray*}$

Now, we consider the set of unit elements of $Z_{m}$ . The $x^{2}\equiv a$ $\; (\text{mod}\; 2)$ has one solution and $x^{2}\equiv a \; (\text{mod}\; 2^{2})$ has two solution. By Theorem 1.2, if $\beta \geq 3,$ the congruence $x^{2}\equiv a \; (\text{mod}\; 2^{\beta})$ has either no solution or exactly 4 incongruent solutions. Furthermore, again by Theorem 1.1, for an odd prime the congruence $x^{2}\equiv a \; (\text{mod}\; q)$ has either no solution or exactly 2 incongruent solutions. By using Chinese remainder theorem, if $\beta = 0\; \text{or}\; 1$ then, $x^{2}\equiv a \; (\text{mod}\; 2^{\beta})$ has either no solution or exactly 2 incongruent solutions. If $\beta = 2$ then, $x^{2}\equiv a \; (\text{mod}\; 2^{\beta})$ has either no solution or exactly 4 incongruent solutions. Lastly, if $\beta \geq 3$ , then $x^{2}\equiv a\; (\text{mod}\; 2^{\beta})$ has either no solution or exactly 8 incongruent solutions. Hence, $\overbrace{G}(2, 2^{\beta}q) = \frac{\phi(n)}{8}K_{8}, \beta\geq3$ . We achieve the desired outcome by combining both cases.

Proposition 3.2. If $q$ and $p$ are odd primes, then $\overbrace{G}(2, qp) = K_{1}\bigoplus\frac{q+p-2}{2}K_{2}\bigoplus \frac{(q-1)(p-1)}{4}K_{4}.$

Proof. To prove this first we discuss the zero-divisors elements of ring $Z_{m}.$ Let $\{ 0, nq, yp | n = 1, 2, 3, \cdots, p-1, y = 1, 2, 3, \cdots, q-1\}$ be a set of zero-divisors of $Z_{qp}$ with zero. There are $\frac{q+p}{2}$ distinct congruences. We have

$\begin{eqnarray} \eta^{2} &\equiv& 0\; (\text{mod} \; qp), \end{eqnarray}$

(3.73)

$\begin{eqnarray} \eta^{2} &\equiv& (qt)^{2}\; (\text{mod} \; qp), \; \; t = 1, 2, 3, \cdots, \frac{p-1}{2}, \end{eqnarray}$

(3.74)

$\begin{eqnarray} \eta^{2} &\equiv& (pt)^{2}\; (\text{mod} \; qp), \; \; t = 1, 2, 3, \cdots, \frac{q-1}{2}. \end{eqnarray}$

(3.75)

Zeroes of these congruences are $\eta = 0$ , $\eta = qt, qp-qt, \; \; t = 1, 2, 3, \cdots, \frac{p-1}{2},$ and $\eta = pt, qp-pt, \; \; t = 1, 2, 3, \cdots, \frac{q-1}{2}.$ Thus, $\overbrace{G}(2, qp) = K_{1}\bigoplus\frac{q+p-2}{2}K_{2}.$ Now we discuss the unit elements of $Z_{m}$ . By Theorem 1.1, for distinct odd primes $q$ and $p$ , the congruence $x^{2}\equiv a \; (\text{mod}\; q)$ has either no solution or exactly 2 incongruent solutions. Similarly, the congruence $x^{2}\equiv a \; (\text{mod}\; p)$ has either no solution or exactly 2 incongruent solutions. By using Chines remainder theorem we have, $x^{2}\equiv a \; (\text{mod}\; q^\beta)$ has either no solution or exactly 4 solutions. Thus, $\overbrace{G}(2, qp) = \frac{\phi(qp)}{4}K_{4} = \frac{\phi(q-1)(p-1))}{4}K_{4}$ . We get the desired outcome by combining both cases.

Theorem 3.3. Let $q$ be an odd prime. Then, $\overbrace{G}(2, 2q^{\beta})$

$= \left\{ \begin{array}{ll} 2K_{1}\bigoplus (q-1)K_{2}, & {\rm\; if\; }\beta = 1, \\ q(q-1)K_{2}\bigoplus2K_{q}, & {\rm\; if\; }\beta = 2, \\ (q-1)q^{2}K_{2}\bigoplus2K_{p}\bigoplus(q-1)K_{2q}, & {\rm\; if\; }\beta = 3, \\ (q-1)q^{\beta-1}~~K_{2}\bigoplus\limits_{i = 1}^{\frac{\beta -2}{2}}q^{\beta-1-2i}(q-1)K_{2q^{i}}\bigoplus 2K_{q^{\beta/2}}, & {\rm\; if\; }\beta\geq 4, {\rm\; and\; }\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \beta\equiv 0\; (\text{mod}\; 2), \\ (q-1)q^{\beta-1}~~K_{2}\bigoplus\limits_{i = 1}^{\frac{\beta -1}{2}}q^{\beta-1-2i}(q-1)K_{2q^{i}}\bigoplus 2K_{q^{(\beta-1)/2}}, & {\rm\; if\; }\beta\geq 5 {\rm\; and\; }\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \beta\equiv 1\; (\text{mod}\; 2).\\ \end{array} \right.$

Proof. The proof is on similar lines as illustrated in the proof of Theorem 3.1.

Figures 4 and 5 reflect Theorems 3.1 and 3.3, respectively.

Figure 4.

$\widetilde{G}(2, 96) = 6K_{4}\bigoplus 7K_{8}\bigoplus K_{16}$ .

DownLoad: Full-Size Img PowerPoint

Figure 5.

$\widetilde{G}(2, 486) = 2K_{18}\bigoplus 2K_{9}\bigoplus 18K_{6}\bigoplus 162K_{2}$ .

DownLoad: Full-Size Img PowerPoint

4. Conclusions

In this article, we investigated the mapping $x^{\alpha}\equiv y^{\alpha}, \; \; (\text{mod}\; m)$ for $\alpha = 2$ over the ring of integers. A problem of partitions of a given set into the form of subsets with equal sums is NP problem. A paradigmatic approach was introduced to find equal sum partitions of quadratic maps via complete graphs. Moreover, we characterized quadratic graphs associated with the mapping $x^{\alpha}\equiv y^{\alpha}, \; \; (\text{mod}\; m),$ $\alpha = 2$ for well-known classes $m = 2^{\beta}, q^{\beta}, 2^{\beta}q, 2q^{\beta}, qp,$ in terms of complete graphs, where $q, p$ is an odd prime. Later on, we intend to extend our research to higher values of $\alpha$ over various rings. We hope that this work will open new inquiry opportunities in various fields for other researchers and knowledge seekers.

Acknowledgment

This work was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of the research article.

Acknowledgments

The authors wish to thank all the researchers who contributed to the research field around the globe.

Conflict of interest

The authors declare no conflicts of interest.

References

[1]	Blagosklonny MV (2005) Teratogens as anti-cancer drugs. Cell Cycle 4: 1518-1521. https://doi.org/10.4161/cc.4.11.2208
[2]	Ramazani M, Jaktaji RP, Shirazi FH, et al. (2019) Analysis of apoptosis related genes in nurses exposed to anti-neoplastic drugs. BMC Pharmacol Toxicol 20. https://doi.org/10.1186/s40360-019-0372-0
[3]	Finnell RH (1999) Teratology: general considerations and principles. J Allergy Clin Immunol 103: S337-S342. https://doi.org/10.1016/S0091-6749(99)70259-9
[4]	Vargesson N (2015) Thalidomide-induced teratogenesis: History and mechanisms. Birth Defects Res Part C Embryo Today Rev 105: 140-156. https://doi.org/10.1002/bdrc.21096
[5]	Coyle I, Wayner MJ, Singer G (1976) Behavioral teratogenesis: A critical evaluation. Pharmacol Biochem Behav 4: 191-200. https://doi.org/10.1016/0091-3057(76)90014-9
[6]	Flaxman SM, Sherman PW (2000) Morning sickness: a mechanism for protecting mother and embryo. Q Rev Biol 75: 113-148. https://doi.org/10.1086/393377
[7]	Mei N, Guo X, Ren Z, et al. (2017) Review of Ginkgo biloba-induced toxicity, from experimental studies to human case reports. J Environ Sci Heal Part C 35: 1-28. https://doi.org/10.1080/10590501.2016.1278298
[8]	Wu C-P, Ohnuma S, V Ambudkar S (2011) Discovering natural product modulators to overcome multidrug resistance in cancer chemotherapy. Curr Pharm Biotechnol 12: 609-620. https://doi.org/10.2174/138920111795163887
[9]	Sung H, Ferlay J, Siegel RL, et al. (2021) Global cancer statistics 2020: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries. CA Cancer J Clin 71: 209-249. https://doi.org/10.3322/caac.21660
[10]	Roy M, Datta A (2019) Cancer: Types and Hallmarks. Cancer Genetics and Therapeutics . Springer 1-26. https://doi.org/10.1007/978-981-13-9471-3_1
[11]	Manzo G (2019) Similarities Between Embryo Development and Cancer Process Suggest New Strategies for Research and Therapy of Tumors: A New Point of View. Front cell Dev Biol 7: 20. https://doi.org/10.3389/fcell.2019.00020
[12]	Shabo I, Svanvik J, Lindström A, et al. (2020) Roles of cell fusion, hybridization and polyploid cell formation in cancer metastasis. World J Clin Oncol 11: 121. https://doi.org/10.5306/wjco.v11.i3.121
[13]	Ledbetter DH (2009) Chaos in the embryo. Nat Med 15: 490-491. https://doi.org/10.1038/nm0509-490
[14]	Erenpreisa J, Giuliani A (2020) Resolution of Complex Issues in Genome Regulation and Cancer Requires Non-Linear and Network-Based Thermodynamics. Int J Mol Sci 21. https://doi.org/10.3390/ijms21010240
[15]	Erenpreisa J, Salmina K, Anatskaya O, et al. (2020) Paradoxes of cancer: Survival at the brink. Semin Cancer Biol . https://doi.org/10.1016/j.semcancer.2020.12.009
[16]	Krebs ET (1947) CANCER AND THE EMBRYONAL HYPOTHESIS. Calif Med 66: 270-271.
[17]	Saitoh M (2018) Involvement of partial EMT in cancer progression. J Biochem 164: 257-264. https://doi.org/10.1093/jb/mvy047
[18]	Saitoh M (2018) JB special review-cellular plasticity in epithelial homeostasis and diseases: Involvement of partial EMT in cancer progression. J Biochem 164: 257-264. https://doi.org/10.1093/jb/mvy047
[19]	Mukund K, Syulyukina N, Ramamoorthy S, et al. (2020) Right and left-sided colon cancers-specificity of molecular mechanisms in tumorigenesis and progression. BMC Cancer 20. https://doi.org/10.1186/s12885-020-06784-7
[20]	Wessely A, Steeb T, Berking C, et al. (2021) How neural crest transcription factors contribute to melanoma heterogeneity, cellular plasticity, and treatment resistance. Int J Mol Sci 22. https://doi.org/10.3390/ijms22115761
[21]	Saravanan KM, Kannan M, Meera P, et al. (2022) E3 ligases: a potential multi-drug target for different types of cancers and neurological disorders. Future Med Chem 14: 187-201. https://doi.org/10.4155/fmc-2021-0157
[22]	Ben-Porath I, Thomson MW, Carey VJ, et al. (2008) An embryonic stem cell-like gene expression signature in poorly differentiated aggressive human tumors. Nat Genet 40: 499-507. https://doi.org/10.1038/ng.127
[23]	Viswanathan A, Musa A, Murugesan A, et al. (2019) Battling Glioblastoma: A Novel Tyrosine Kinase Inhibitor with Multi-Dimensional Anti-Tumor Effect (Running Title: Cancer Cells Death Signalling Activation). Cells 8. https://doi.org/10.3390/cells8121624
[24]	Doan P, Nguyen P, Murugesan A, et al. (2021) Targeting orphan g protein-coupled receptor 17 with t0 ligand impairs glioblastoma growth. Cancers (Basel) 13. https://doi.org/10.3390/cancers13153773
[25]	Cofre J, Abdelhay E (2017) Cancer Is to Embryology as Mutation Is to Genetics: Hypothesis of the Cancer as Embryological Phenomenon. ScientificWorldJournal 2017: 3578090. https://doi.org/10.1155/2017/3578090
[26]	Rountree CB, Mishra L, Willenbring H (2012) Stem cells in liver diseases and cancer: recent advances on the path to new therapies. Hepatology 55: 298-306. https://doi.org/10.1002/hep.24762
[27]	Qin S, Jiang J, Lu Y, et al. (2020) Emerging role of tumor cell plasticity in modifying therapeutic response. Signal Transduct Target Ther 5: 228. https://doi.org/10.1038/s41392-020-00313-5
[28]	Pierce GB (1983) The cancer cell and its control by the embryo. Rous-Whipple Award lecture. Am J Pathol 113: 117.
[29]	Liu J (2018) The dualistic origin of human tumors. Semin Cancer Biol 53: 1-16. https://doi.org/10.1016/j.semcancer.2018.07.004
[30]	Xu B, Konze KD, Jin J, et al. (2015) Targeting EZH2 and PRC2 dependence as novel anticancer therapy. Exp Hematol 43: 698-712. https://doi.org/10.1016/j.exphem.2015.05.001
[31]	Wang W, Qin J, Voruganti S, et al. (2015) Polycomb group (PcG) proteins and human cancers: multifaceted functions and therapeutic implications. Med Res Rev 35: 1220-1267. https://doi.org/10.1002/med.21358
[32]	Murray MJ, Lessey BA (1999) Embryo implantation and tumor metastasis: Common pathways of invasion and angiogenesis. Semin Reprod Endocrinol 17: 275-290. https://doi.org/10.1055/s-2007-1016235
[33]	Costanzo V, Bardelli A, Siena S, et al. (2018) Exploring the links between cancer and placenta development. Open Biol 8: 180081. https://doi.org/10.1098/rsob.180081
[34]	Turajlic S, Sottoriva A, Graham T, et al. (2019) Resolving genetic heterogeneity in cancer. Nat Rev Genet 20: 404-416. https://doi.org/10.1038/s41576-019-0114-6
[35]	Wagner GP, Kshitiz, Dighe A, et al. (2021) The Coevolution of Placentation and Cancer. Annu Rev Anim Biosci 10. https://doi.org/10.1146/annurev-animal-020420-031544
[36]	Fukuda MN, Sugihara K (2008) An integrated view of L-selectin and trophinin function in human embryo implantation. J Obstet Gynaecol Res 34: 129-136. https://doi.org/10.1111/j.1447-0756.2008.00776.x
[37]	Ayala GE, Dai H, Li R, et al. (2006) Bystin in perineural invasion of prostate cancer. Prostate 66: 266-272. https://doi.org/10.1002/pros.20323
[38]	Aplin JD, Ruane PT (2017) Embryo–epithelium interactions during implantation at a glance. J Cell Sci 130: 15-22. https://doi.org/10.1242/jcs.175943
[39]	Harada O, Suga T, Suzuki T, et al. (2007) The role of trophinin, an adhesion molecule unique to human trophoblasts, in progression of colorectal cancer. Int J Cancer 121: 1072-1078. https://doi.org/10.1002/ijc.22821
[40]	Sims GP, Rowe DC, Rietdijk ST, et al. (2009) HMGB1 and RAGE in inflammation and cancer. Annu Rev Immunol 28: 367-388. https://doi.org/10.1146/annurev.immunol.021908.132603
[41]	Ibrahim ZA, Armour CL, Phipps S, et al. (2013) RAGE and TLRs: relatives, friends or neighbours?. Mol Immunol 56: 739-744. https://doi.org/10.1016/j.molimm.2013.07.008
[42]	Cui T, Zhang W, Li S, et al. (2019) Oxidative Stress–Induced HMGB1 Release from Melanocytes: A Paracrine Mechanism Underlying the Cutaneous Inflammation in Vitiligo. J Invest Dermatol 139: 2174-2184.e4. https://doi.org/10.1016/j.jid.2019.03.1148
[43]	Rapoport BL, Steel HC, Theron AJ, et al. (2020) High Mobility Group Box 1 in Human Cancer. Cells 9. https://doi.org/10.3390/cells9071664
[44]	Zhang H, Li J, Saravanan KM, et al. (2021) An Integrated Deep Learning and Molecular Dynamics Simulation-Based Screening Pipeline Identifies Inhibitors of a New Cancer Drug Target TIPE2. Front Pharmacol 12. https://doi.org/10.3389/fphar.2021.772296
[45]	Saravanan KM, Zhang H, Zhang H, et al. (2020) On the Conformational Dynamics of β-Amyloid Forming Peptides: A Computational Perspective. Front Bioeng Biotechnol 8. https://doi.org/10.3389/fbioe.2020.00532
[46]	Viswanathan A, Kute D, Musa A, et al. (2019) 2-(2-(2,4-dioxopentan-3-ylidene)hydrazineyl)benzonitrile as novel inhibitor of receptor tyrosine kinase and PI3K/AKT/mTOR signaling pathway in glioblastoma. Eur J Med Chem 166: 291-303. https://doi.org/10.1016/j.ejmech.2019.01.021
[47]	Imashiro C, Azuma T, Itai S, et al. (2021) Travelling ultrasound promotes vasculogenesis of three-dimensional-monocultured human umbilical vein endothelial cells. Biotechnol Bioeng 118: 3760-3769. https://doi.org/10.1002/bit.27852
[48]	Srimathi Devi J, Haripriya D, Arul S, et al. (2021) Evaluation of anti-cancer effect of zerumbone and cisplatin on N-nitrosodiethylamine induced hepatic cancer in freshwater fish (Danio rerio). Nat Prod Res 1–5. https://doi.org/10.1080/14786419.2021.2012672
[49]	Saravanan KM, Ponnuraj K (2018) Sequence and structural analysis of fibronectin-binding protein reveals importance of multiple intrinsic disordered tandem repeats. J Mol Recognit : e2768. https://doi.org/10.1002/jmr.2768
[50]	George EL, Georges-Labouesse EN, Patel-King RS, et al. (1993) Defects in mesoderm, neural tube and vascular development in mouse embryos lacking fibronectin. Development 119: 1079-1091. https://doi.org/10.1242/dev.119.4.1079
[51]	Snow CJ, Goody M, Kelly MW, et al. (2008) Time-lapse analysis and mathematical characterization elucidate novel mechanisms underlying muscle morphogenesis. PLoS Genet 4. https://doi.org/10.1371/journal.pgen.1000219
[52]	Murphy PA, Begum S, Hynes RO (2015) Tumor angiogenesis in the absence of fibronectin or its cognate integrin receptors. PLoS One 10. https://doi.org/10.1371/journal.pone.0120872
[53]	Galmiche A, Rak J, Roumenina LT, et al. (2022) Coagulome and the tumor microenvironment: an actionable interplay. Trends in Cancer . https://doi.org/10.1016/j.trecan.2021.12.008
[54]	Nasser NJ, Fox J, Agbarya A (2020) Potential mechanisms of cancer-related hypercoagulability. Cancers (Basel) 12. https://doi.org/10.3390/cancers12030566
[55]	Yu G, Wen Q (2018) Expression of embryonic liver fodrin (ELF) and stem cell markers in CD13 liver cancer stem cells. Eur Rev Med Pharmacol Sci 22: 1653-1657.
[56]	Baek HJ, Lim SC, Kitisin K, et al. (2008) Hepatocellular cancer arises from loss of transforming growth factor beta signaling adaptor protein embryonic liver fodrin through abnormal angiogenesis. Hepatology 48: 1128-1137. https://doi.org/10.1002/hep.22460
[57]	Teng B, Huang C, Cheng CL, et al. (2020) Newly identified peptide hormone inhibits intestinal fat absorption and improves NAFLD through its receptor GPRC6A. J Hepatol 73: 383-393. https://doi.org/10.1016/j.jhep.2020.02.026
[58]	Bian J, Dannappel M, Wan C, et al. (2020) Transcriptional regulation of Wnt/β-catenin pathway in colorectal cancer. Cells 9: 2125. https://doi.org/10.3390/cells9092125
[59]	Davis-Marcisak EF, Deshpande A, Stein-O'Brien GL, et al. (2021) From bench to bedside: Single-cell analysis for cancer immunotherapy. Cancer Cell 39: 1062-1080. https://doi.org/10.1016/j.ccell.2021.07.004
[60]	Xie Y, Yao L, Yu X, et al. (2020) Action mechanisms and research methods of tRNA-derived small RNAs. Signal Transduct Target Ther 5: 1-9. https://doi.org/10.1038/s41392-020-00217-4
[61]	Postovit LM, Margaryan NV, Seftor EA, et al. (2008) Human embryonic stem cell microenvironment suppresses the tumorigenic phenotype of aggressive cancer cells. Proc Natl Acad Sci U S A 105: 4329-4334. https://doi.org/10.1073/pnas.0800467105
[62]	Dreesen O, Brivanlou AH (2007) Signaling pathways in cancer and embryonic stem cells. Stem Cell Rev 3: 7-17. https://doi.org/10.1007/s12015-007-0004-8
[63]	Saravanan KM, Palanivel S, Yli-Harja O, et al. (2018) Identification of novel GPR17-agonists by structural bioinformatics and signaling activation. Int J Biol Macromol 106: 901-907. https://doi.org/10.1016/j.ijbiomac.2017.08.088
[64]	Caruso S, O'Brien DR, Cleary SP, et al. (2021) Genetics of hepatocellular carcinoma: approaches to explore molecular diversity. Hepatology 73: 14-26. https://doi.org/10.1002/hep.31394
[65]	Gnanavel M, Murugesan A, Mani SK, et al. (2021) Identifying the mirna signature association with aging-related senescence in glioblastoma. Int J Mol Sci 22: 1-14. https://doi.org/10.3390/ijms22020517
[66]	Buhrmann C, Brockmueller A, Harsha C, et al. (2021) Evidence that tumor microenvironment initiates epithelial-to-mesenchymal transition and calebin a can suppress it in colorectal cancer cells. Front Pharmacol 12: 1689. https://doi.org/10.3389/fphar.2021.699842
[67]	Zhang H, Shao X, Peng Y, et al. (2019) A novel machine learning based approach for iPS progenitor cell identification. PLoS Comput Biol . https://doi.org/10.1101/744920
[68]	Shetzer Y, Solomon H, Koifman G, et al. (2014) The paradigm of mutant p53-expressing cancer stem cells and drug resistance. Carcinogenesis 35: 1196-1208. https://doi.org/10.1093/carcin/bgu073
[69]	Filippi L, Pini A, Cammalleri M, et al. (2021) β3-Adrenoceptor, a novel player in the round-trip from neonatal diseases to cancer: Suggestive clues from embryo. Med Res Rev : n/a. https://doi.org/10.1002/med.21874
[70]	Martinez NJ, Walhout AJM (2009) The interplay between transcription factors and microRNAs in genome-scale regulatory networks. Bioessays 31: 435-445. https://doi.org/10.1002/bies.200800212
[71]	Shi G, Jin Y (2010) Role of Oct4 in maintaining and regaining stem cell pluripotency. Stem Cell Res Ther 1: 1-9. https://doi.org/10.1186/scrt39
[72]	van den Berg DLC, Snoek T, Mullin NP, et al. (2010) An Oct4-centered protein interaction network in embryonic stem cells. Cell Stem Cell 6: 369-381. https://doi.org/10.1016/j.stem.2010.02.014
[73]	Innes KE, Byers TE (1999) Preeclampsia and breast cancer risk. Epidemiology 10: 722-732. https://doi.org/10.1097/00001648-199911000-00013
[74]	Mahendra G, Kumar S, Isayeva T, et al. (2005) Antiangiogenic cancer gene therapy by adeno-associated virus 2-mediated stable expression of the soluble FMS-like tyrosine kinase-1 receptor. Cancer Gene Ther 12: 26-34. https://doi.org/10.1038/sj.cgt.7700754
[75]	Bellamy L, Casas JP, Hingorani AD, et al. (2007) Pre-eclampsia and risk of cardiovascular disease and cancer in later life: Systematic review and meta-analysis. Br Med J 335: 974-977. https://doi.org/10.1136/bmj.39335.385301.BE
[76]	Wu P, Haththotuwa R, Kwok CS, et al. (2017) Preeclampsia and future cardiovascular health: a systematic review and meta-analysis. Circ Cardiovasc Qual Outcomes 10: e003497. https://doi.org/10.1161/CIRCOUTCOMES.116.003497
[77]	Fong GH, Rossant J, Gertsenstein M, et al. (1995) Role of the Flt-1 receptor tyrosine kinase in regulating the assembly of vascular endothelium. Nature 376: 66-70. https://doi.org/10.1038/376066a0
[78]	Shibuya M, Yamaguchi S, Yamane A, et al. (1990) Nucleotide sequence and expression of a novel human receptor-type tyrosine kinase gene (flt) closely relatd to the fms family. Oncogene 5: 519-524.
[79]	Kim JS, Kang EJ, Woo OH, et al. (2013) The relationship between preeclampsia, pregnancy-induced hypertension and maternal risk of breast cancer: A meta-analysis. Acta Oncol (Madr) 52: 1643-1648. https://doi.org/10.3109/0284186X.2012.750033
[80]	Calderon-Margalit R, Friedlander Y, Yanetz R, et al. (2009) Preeclampsia and subsequent risk of cancer: update from the Jerusalem Perinatal Study. Am J Obstet Gynecol 200: 63.e1-63.e5. https://doi.org/10.1016/j.ajog.2008.06.057
[81]	Yoshitomi Y, Ikeda T, Saito H, et al. (2017) JunB regulates angiogenesis and neurovascular parallel alignment in mouse embryonic skin. J Cell Sci 130: 916-926. https://doi.org/10.1242/jcs.196303
[82]	Mihajlović AI, FitzHarris G (2018) Segregating chromosomes in the mammalian oocyte. Curr Biol 28: R895-R907. https://doi.org/10.1016/j.cub.2018.06.057
[83]	Wang J, Batourina E, Schneider K, et al. (2018) Polyploid superficial cells that maintain the urothelial barrier are produced via incomplete cytokinesis and endoreplication. Cell Rep 25: 464-477. https://doi.org/10.1016/j.celrep.2018.09.042
[84]	Zhang J, Qiao Q, Xu H, et al. (2021) Human cell polyploidization: the good and the evil. Seminars in Cancer Biology . Elsevier. https://doi.org/10.1016/j.semcancer.2021.04.005
[85]	Vasudevan A, Schukken KM, Sausville EL, et al. (2021) Aneuploidy as a promoter and suppressor of malignant growth. Nat Rev Cancer 21: 89-103. https://doi.org/10.1038/s41568-020-00321-1
[86]	Zhu J, Tsai H-J, Gordon MR, et al. (2018) Cellular stress associated with aneuploidy. Dev Cell 44: 420-431. https://doi.org/10.1016/j.devcel.2018.02.002
[87]	Vázquez-Diez C, FitzHarris G (2018) Causes and consequences of chromosome segregation error in preimplantation embryos. Reproduction 155: R63-R76. https://doi.org/10.1530/REP-17-0569
[88]	Masset H, Tšuiko O, Vermeesch JR (2021) Genome-wide abnormalities in embryos: Origins and clinical consequences. Prenat Diagn 41: 554-563. https://doi.org/10.1002/pd.5895
[89]	Sikora E, Czarnecka-Herok J, Bojko A, et al. (2020) Therapy-induced polyploidization and senescence: Coincidence or interconnection?. Seminars in Cancer Biology . Elsevier. https://doi.org/10.1016/j.semcancer.2020.11.015
[90]	Wang N, Hao F, Shi Y, et al. (2021) The Controversial Role of Polyploidy in Hepatocellular Carcinoma. Onco Targets Ther 14: 5335. https://doi.org/10.2147/OTT.S340435
[91]	Amend SR, Torga G, Lin K, et al. (2019) Polyploid giant cancer cells: Unrecognized actuators of tumorigenesis, metastasis, and resistance. Prostate 79: 1489-1497. https://doi.org/10.1002/pros.23877
[92]	Was H, Borkowska A, Olszewska A, et al. (2021) Polyploidy formation in cancer cells: How a Trojan horse is born. Seminars in Cancer Biology . Elsevier. https://doi.org/10.1016/j.semcancer.2021.03.003
[93]	Demin S, Berdieva M, Goodkov A, et al. (2021) Cell-cell fusions and cell-in-cell phenomena in healthy cells and cancer: Lessons from protists and invertebrates. Seminars in Cancer Biology . Elsevier. https://doi.org/10.1016/j.semcancer.2021.03.005
[94]	Zack TI, Schumacher SE, Carter SL, et al. (2013) Pan-cancer patterns of somatic copy number alteration. Nat Genet 45: 1134-1140. https://doi.org/10.1038/ng.2760
[95]	Chen J, Niu N, Zhang J, et al. (2019) Polyploid giant cancer cells (PGCCs): the evil roots of cancer. Curr Cancer Drug Targets 19: 360-367. https://doi.org/10.2174/1568009618666180703154233
[96]	Wen Q, Goldenson B, Silver SJ, et al. (2012) Identification of regulators of polyploidization presents therapeutic targets for treatment of AMKL. Cell 150: 575-589. https://doi.org/10.1016/j.cell.2012.06.032
[97]	Vakifahmetoglu H, Olsson M, Zhivotovsky B (2008) Death through a tragedy: mitotic catastrophe. Cell Death Differ 15: 1153-1162. https://doi.org/10.1038/cdd.2008.47
[98]	Ganem NJ, Storchova Z, Pellman D (2007) Tetraploidy, aneuploidy and cancer. Curr Opin Genet Dev 17: 157-162. https://doi.org/10.1016/j.gde.2007.02.011
[99]	Ganem NJ, Pellman D (2007) Limiting the proliferation of polyploid cells. Cell 131: 437-440. https://doi.org/10.1016/j.cell.2007.10.024
[100]	Leikam C, Hufnagel AL, Otto C, et al. (2015) In vitro evidence for senescent multinucleated melanocytes as a source for tumor-initiating cells. Cell Death Dis 6: e1711-e1711. https://doi.org/10.1038/cddis.2015.71
[101]	Zheng L, Dai H, Zhou M, et al. (2012) Polyploid cells rewire DNA damage response networks to overcome replication stress-induced barriers for tumour progression. Nat Commun 3: 1-12. https://doi.org/10.1038/ncomms1825
[102]	Thura M, Ye Z, Al-Aidaroos AQ, et al. (2021) PRL3 induces polyploid giant cancer cells eliminated by PRL3-zumab to reduce tumor relapse. Commun Biol 4: 923. https://doi.org/10.1038/s42003-021-02449-8
[103]	Liu J (2020) The “life code”: A theory that unifies the human life cycle and the origin of human tumors. Semin Cancer Biol 60: 380-397. https://doi.org/10.1016/j.semcancer.2019.09.005
[104]	Niu N, Yao J, Bast RC, et al. (2021) IL-6 promotes drug resistance through formation of polyploid giant cancer cells and stromal fibroblast reprogramming. Oncogenesis 10: 65. https://doi.org/10.1038/s41389-021-00349-4
[105]	Campbell AM, Zhang Z-Y (2014) Phosphatase of regenerating liver: a novel target for cancer therapy. Expert Opin Ther Targets 18: 555-569. https://doi.org/10.1517/14728222.2014.892926
[106]	Shankaranarayanan JS, Kanwar JR, AL-Juhaishi AJA, et al. (2016) Doxorubicin Conjugated to Immunomodulatory Anticancer Lactoferrin Displays Improved Cytotoxicity Overcoming Prostate Cancer Chemo resistance and Inhibits Tumour Development in TRAMP Mice. Sci Rep 6: 32062. https://doi.org/10.1038/srep32062
[107]	Niu N, Mercado-Uribe I, Liu J (2017) Dedifferentiation into blastomere-like cancer stem cells via formation of polyploid giant cancer cells. Oncogene 36: 4887-4900. https://doi.org/10.1038/onc.2017.72
[108]	Xiong S, Feng Y, Cheng L (2019) Cellular reprogramming as a therapeutic target in cancer. Trends Cell Biol 29: 623-634. https://doi.org/10.1016/j.tcb.2019.05.001
[109]	Labi V, Erlacher M (2015) How cell death shapes cancer. Cell Death Dis 6: e1675-e1675. https://doi.org/10.1038/cddis.2015.20
[110]	Zhang D, Yang X, Yang Z, et al. (2017) Daughter Cells and Erythroid Cells Budding from PGCCs and Their Clinicopathological Significances in Colorectal Cancer. J Cancer 8: 469-478. https://doi.org/10.7150/jca.17012
[111]	Zhang S, Mercado-Uribe I, Hanash S, et al. (2013) iTRAQ-based proteomic analysis of polyploid giant cancer cells and budding progeny cells reveals several distinct pathways for ovarian cancer development. PLoS One 8: e80120. https://doi.org/10.1371/journal.pone.0080120
[112]	Lv H, Shi Y, Zhang L, et al. (2014) Polyploid giant cancer cells with budding and the expression of cyclin E, S-phase kinase-associated protein 2, stathmin associated with the grading and metastasis in serous ovarian tumor. BMC Cancer 14: 576. https://doi.org/10.1186/1471-2407-14-576
[113]	Xu J, Hardin H, Zhang R, et al. (2016) Stage-Specific Embryonic Antigen-1 (SSEA-1) Expression in Thyroid Tissues. Endocr Pathol 27: 271-275. https://doi.org/10.1007/s12022-016-9448-1
[114]	Fleming H (2021) Opaque Polyploid Cells in Ishikawa Endometrial Cultures Are Capable of Forming Megamitochondria, Organelles Derived from the Adaptation of Fused Mitochondria Whose Capacity to Develop Gaseous Vacuoles Suggests CO2 Retention and Hypoxic Metabolism. Adv Biosci Biotechnol 12: 229-255. https://doi.org/10.4236/abb.2021.127015
[115]	Liu J (2021) Giant cells: Linking McClintock's heredity to early embryogenesis and tumor origin throughout millennia of evolution on Earth. Seminars in Cancer Biology . Elsevier. https://doi.org/10.1016/j.semcancer.2021.06.007
[116]	Liu J, Niu N, Li X, et al. (2021) The life cycle of polyploid giant cancer cells and dormancy in cancer: Opportunities for novel therapeutic interventions. Semin Cancer Biol . https://doi.org/10.1016/j.semcancer.2021.10.005
[117]	Li Z, Zheng M, Zhang H, et al. (2021) Arsenic Trioxide Promotes Tumor Progression by Inducing the Formation of PGCCs and Embryonic Hemoglobin in Colon Cancer Cells. Front Oncol 4046. https://doi.org/10.3389/fonc.2021.720814
[118]	Jang YJ, Won JH, Back MJ, et al. (2015) Paraquat induces apoptosis through a mitochondria-dependent pathway in RAW264.7 cells. Biomol Ther 23: 407-413. https://doi.org/10.4062/biomolther.2015.075
[119]	Smith DG, Sturmey RG (2013) Parallels between embryo and cancer cell metabolism. Biochem Soc Trans 41: 664-669. https://doi.org/10.1042/BST20120352
[120]	Christensen DR, Calder PC, Houghton FD (2015) GLUT3 and PKM2 regulate OCT4 expression and support the hypoxic culture of human embryonic stem cells. Sci Rep 5. https://doi.org/10.1038/srep17500
[121]	Dang CV (2013) MYC, metabolism, cell growth, and tumorigenesis. Cold Spring Harb Perspect Med 3. https://doi.org/10.1101/cshperspect.a014217
[122]	Varlakhanova NV, Cotterman RF, deVries WN, et al. (2010) Myc maintains embryonic stem cell pluripotency and self-renewal. Differentiation 80: 9-19. https://doi.org/10.1016/j.diff.2010.05.001
[123]	Gabryelska A, Karuga FF, Szmyd B, et al. (2020) HIF-1α as a Mediator of Insulin Resistance, T2DM, and Its Complications: Potential Links With Obstructive Sleep Apnea. Front Physiol 11. https://doi.org/10.3389/fphys.2020.01035
[124]	Yu M, Lun J, Zhang H, et al. (2021) The non-canonical functions of HIF prolyl hydroxylases and their dual roles in cancer. Int J Biochem Cell Biol 135. https://doi.org/10.1016/j.biocel.2021.105982
[125]	Kim J, Hong J, Lee J, et al. (2021) Recent advances in tumor microenvironment-targeted nanomedicine delivery approaches to overcome limitations of immune checkpoint blockade-based immunotherapy. J Control Release 332: 109-126. https://doi.org/10.1016/j.jconrel.2021.02.002
[126]	Parsons MJ, Tammela T, Dow LE (2021) WNT as a driver and dependency in cancer. Cancer Discov 11: 2413-2429. https://doi.org/10.1158/2159-8290.CD-21-0190
[127]	Wang X, Wang Q (2018) Alpha-fetoprotein and hepatocellular carcinoma immunity. Can J Gastroenterol Hepatol 2018. https://doi.org/10.1155/2018/9049252
[128]	Vujanovic L, Stahl EC, Pardee AD, et al. (2017) Tumor-derived a-fetoprotein directly drives human natural killer-cell activation and subsequent cell death. Cancer Immunol Res 5: 493-502. https://doi.org/10.1158/2326-6066.CIR-16-0216
[129]	Samet JM, Avila-Tang E, Boffetta P, et al. (2009) Lung cancer in never smokers: Clinical epidemiology and environmental risk factors. Clin Cancer Res 15: 5626-5645. https://doi.org/10.1158/1078-0432.CCR-09-0376
[130]	Boogerd LSF, Handgraaf HJM, Lam HD, et al. (2017) Laparoscopic detection and resection of occult liver tumors of multiple cancer types using real-time near-infrared fluorescence guidance. Surg Endosc 31: 952-961. https://doi.org/10.1007/s00464-016-5007-6
[131]	Samanta S, Guru S, Elaimy AL, et al. (2018) IMP3 Stabilization of WNT5B mRNA Facilitates TAZ Activation in Breast Cancer. Cell Rep 23: 2559-2567. https://doi.org/10.1016/j.celrep.2018.04.113
[132]	Beilerli A, Gareev I, Beylerli O, et al. (2021) Circular RNAs as biomarkers and therapeutic targets in cancer. Seminars in Cancer Biology . Elsevier.
[133]	Zhu T, Tu SZ, Yang YL, et al. (2019) Induced effects of resveratrol on apoptosis and autophagy in human breast cancer MDA-MB231 cells. Chinese Pharmacol Bull 35: 839-843. https://doi.org/10.1016/j.semcancer.2020.12.026
[134]	Wu Y, Wang F, Albert Reece E, et al. (2015) Curcumin ameliorates high glucose-induced neural tube defects by suppressing cellular stress and apoptosis. Am J Obstet Gynecol 212: 802.e1-802.e8. https://doi.org/10.1016/j.ajog.2015.01.017

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2. Quadratic residues graphs over $Z_{2^{\beta}}$ and $Z_{q^{\beta}}$

3. Quadratic residues graphs over $Z_{2^{\beta}q},$ $Z_{qp}$ and $Z_{2q^{\beta}}$

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Instrumentals behind embryo and cancer: a platform for prospective future in cancer research

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2. Quadratic residues graphs over Z2β Z_{2^{\beta}} and Zqβ Z_{q^{\beta}}

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2. Quadratic residues graphs over $Z_{2^{\beta}}$ and $Z_{q^{\beta}}$

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