Epigenetic modulation of human neurobiological disorders: Lesch-Nyhan disease as a model disorder

Khue Vu Nguyen; Khue Vu Nguyen

doi:10.3934/Neuroscience.2025005

AIMS Neuroscience

2025, Volume 12, Issue 2: 58-74. doi: 10.3934/Neuroscience.2025005

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Review

Epigenetic modulation of human neurobiological disorders: Lesch-Nyhan disease as a model disorder

Khue Vu Nguyen ^{1,2
,
,}

1.
School of Medical Imaging, Jiangsu Vocational College of Medicine, Yancheng 224005 Jiangsu, China
2.
Center for Molecular Biophysics, CNRS Orleans, 45071 Orleans, France
Former Institution Attended:

Received: 06 October 2024 Revised: 15 March 2025 Accepted: 07 April 2025 Published: 15 April 2025

Epigenetics is the study of how cells control gene activity without changing the DNA sequence. Epigenetic changes affect how genes are turned on and off or expressed, and thus help regulate how cells in different parts of the body use the same genetic code. Errors in the epigenetic process can not only lead to abnormal gene activity or inactivity, but can also influence alternative splicing (AS) and could cause human diseases. Understanding of how epigenetic defects can affect human health, especially for neurological disorders, could suggest targets for therapeutic interventions. For such a purpose, the Lesch-Nyhan disease (LND) has been selected as a valuable model to study the genetic-epigenetic interplay, especially to explore the epistasis between the housekeeping hypoxanthine phosphoribosyltransferase 1 (HPRT1) and β-amyloid precursor protein (APP) genes. This review is structured as follows: we begin with an overview about the monogenetic neurological disorders associated with epigenetic changes; next, the current knowledge on HPRT1 and APP genes is provided; then, the epistasis between HPRT1 and APP genes related to the neurobehavioral syndrome in LND is described; and finally, we present the construction of expression vectors to study intermolecular interactions between the hypoxanthine-guanine phosphoribosyltransferase (HGprt) enzyme and APP in LND. Information obtained from such expression vectors would be useful for future directions to design therapies through epigenetic interventions.

Keywords:

Citation: Khue Vu Nguyen. Epigenetic modulation of human neurobiological disorders: Lesch-Nyhan disease as a model disorder[J]. AIMS Neuroscience, 2025, 12(2): 58-74. doi: 10.3934/Neuroscience.2025005

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Abstract

Abbreviations

ACE2:

Angiotensin-converting enzyme 2;

acetyl-CoA:

acetyl coenzyme A;

AD:

Alzheimer's disease;

ADAM10:

A disintegrin and metalloprotease-10;

AICD:

APP intracellular domain;

ALS:

Amyotrophic lateral sclerosis;

APP:

β-amyloid precursor protein;

APLP2:

APP-like protein-2;

APPsα:

Soluble APP derivative obtained after cleavage by α-secretase;

APPsβ:

Soluble APP derivative obtained after cleavage by β-secretase;

APP-CTFα:

APP-α-carboxyl-terminal fragment;

APP-CTFβ:

APP-β-carboxyl-terminal fragment;

AS:

Alternative splicing;

Aβ:

Amyloid-β peptide;

A713:

Alanine at position 713;

5-aza-CR:

cytidine analogues 5-azacytidine;

BACE1:

β-site APP cleaving enzyme 1;

BET:

Bromodomain and extraterminal;

CA:

Carbonic anhydrase;

CDS:

entire coding sequence;

COVID-19 virus:

Coronavirus disease 2019 virus;

CNS:

Central nervous system;

DNA:

Deoxyribonucleic acid;

DNMTs:

DNA methyltransferases;

EC:

Extracellular domain;

EHMT1:

Euchromatic histone-lysine N-methyltransferase 1;

FAD:

Familial AD;

FOLR1:

human folate receptor 1;

FXS:

Fragile X syndrome;

GFP:

Green fluorescent protein;

GLP:

G9a-like protein;

GMP:

Guanosine monophosphate;

GPI:

Glycosylphosphatidylinositol;

GPI-APs:

GPI-anchored proteins;

HATs:

Histone acetyltransferases;

HDACs:

Histone deacetylases;

HGprt:

Hypoxanthine-guanine phosphoribosyltransferase;

HLA:

Human leukocyte antigen;

HPRT1:

Hypoxanthine phosphoribosyltransferase 1;

HMTs:

Histone methyltransferases;

IC:

Intracellular domain;

IMP:

Inosine monophosphate;

INDELS:

Deletion followed by an insertion;

KPI:

Kunitz-type serine protease inhibitor;

K687:

Lysine at position 687;

LND:

Lesch-Nyhan disease;

LNVs:

Lessh-Nyhan variants;

MBD:

Methyl-CpG-binding;

MECP2:

Methyl CpG binding protein 2;

MIM:

Mendelian inheritance in man;

mRNA:

Messenger RNA;

MS:

Multiple sclerosis;

M671:

Methionine at position 671;

NFTs:

Neurofibrillary tangles;

NINCDS-ADRDA:

National Institute of Neurological and Communicative Disorders and Stroke and the Alzheimer's disease and Related Disorder Association;

PGD:

phosphogluconate dehydrogenase;

pre-mRNA:

Precursor mRNA;

PrPc:

Cellular prion protein (the ubiquitous normal cellular form);

PRPP:

α-D-5-phosphoribosyl-1-pyrophosphate;

PS:

Presenilin;

RNA:

Ribonucleic acid;

SAD:

Sporadic AD;

SARS-CoV-2:

Severe acute respiratory syndrome coronavirus 2;

S glycoprotein:

Spike glycoprotein;

SNP:

Single-nucleotide polymorphism;

SPs:

Senile plaques;

TM:

Transmembrane domain;

TSEs:

Transmissible spongiform encephalopathies;

V711:

Valine at position 711

1. Introduction

Let $q > 1$ be an integer and let $\chi$ be Dirichlet character modulo $q$ . The Gauss sums $G(m, \chi; q)$ is defined as

$G(m, \chi; q) = \sum\limits_{a = 1}^{q}\chi(a) e\left(\frac{ma}{q}\right),$

where $m$ is integer and $e(y) = e^{2\pi iy}$ .

Gauss sums is very important in the analytic number theory and related research filed. Many scholars studied its properties and obtained a series of interesting results (see [1-4,6,7,9-14,,]). For example, let $\chi$ be the primitive character modulo $q$ , we have

$G(m, \chi;q) = \overline{\chi}(m)G(1, \chi;q)\equiv \overline{\chi}(m)\tau(\chi)\ \ {\rm{and}} \ \ \left|\tau(\chi)\right| = \sqrt{q},$

where $\tau\left(\chi\right) = \sum_{b = 1}^q\chi(b)e\left(\frac{b}{q}\right)$ and $\overline{\chi}$ is the complex conjugate of $\chi$ .

Berndt and Evans [] studied the properties of cubic Gauss sums. Zhang and Hu [] studied the number of the solutions of the diagonal cubic congruence equation mod $p$ , they obtained the following results: Let $p$ be a prime with $p\equiv 1 \bmod 3$ . Then for any third-order character $\lambda$ modulo $p$ , one has the identity

$\begin{eqnarray} \tau^3(\lambda)+ \tau^3\left(\overline{\lambda}\right) = dp, \end{eqnarray}$

(1.1)

where $d$ is uniquely determined by $4p = d^2+27b^2$ and $d\equiv 1\bmod 3$ .

In addition, Chen and Zhang [] studied the case of the fourth-order character modulo $p$ , and obtained an identity (see the Lemma 1). Chen [] studied the properties of the Gauss sums of the sixth-order character modulo $p$ . It is not hard to see from [5,,], the number of these characters is $2$ . What about the number of the characters $> 2$ ? Motivated by that, Zhang et al. [16] studied the eight-order and twelve-order characters.

In this article, we shall further study the generalization. That is, the number of the characters is $\phi\left(2^k\right) = 2^{k-1}\geq 4$ . We prove several identities involving the Gauss sums of the $2^k$ -order character modulo an odd prime $p$ with $p\equiv 1\bmod 2^k$ ( $k\geq 3$ ). We give several exact and interesting formulae for them.

Theorem 1. Let $p$ be an odd prime with $p\equiv 1\bmod 16$ . If $\chi_{16}$ denotes asixteen-order character modulo $p$ , then we have the identity

$\begin{equation*} \left|\frac{\tau^4\left(\chi^7_{16}\right)}{\tau^4\left(\chi_{16}\right)}+\frac{\tau^4\left(\chi^5_{16}\right)}{\tau^4\left(\chi^3_{16}\right)}\right| = \left|\frac{\tau^4\left(\chi^9_{16}\right)}{\tau^4\left(\chi^{15}_{16}\right)}+\frac{\tau^4\left(\chi^{11}_{16}\right)}{\tau^4\left(\chi^{13}_{16}\right)}\right| = \frac{2\cdot |\alpha| }{\sqrt{p}}, \end{equation*}$

where $\alpha = \frac{1}{2}\sum\limits_{a = 1}^{p-1}\left(\frac{a+\bar{a}}{p}\right)$ , $\bar{a}$ denotes the multiplicative inverse of $a$ modulo $p$ , and $\left(\frac{*}{p}\right)$ denotes the Legendre's symbol modulo $p$ .

Theorem 2. Let $p$ be an odd prime with $p\equiv 1\bmod 32$ . If $\chi_{32}$ denotes a $32$ -order character modulo $p$ , then we have the identity

$\begin{equation*} \left|\frac{\tau^4\left(\chi^7_{32}\right)\cdot \tau^4\left(\chi_{32}^{15}\right)}{\tau^4\left(\chi_{32}\right)\cdot\tau^4\left(\chi_{32}^{9}\right) }+ \frac{\tau^4\left(\chi^{5}_{32}\right)\cdot \tau^4\left(\chi_{32}^{13}\right)}{\tau^4\left(\chi_{32}^{3}\right)\cdot\tau^4\left(\chi_{32}^{11}\right) } \right| = \frac{2\cdot |\alpha |}{\sqrt{p}}. \end{equation*}$

More generally, for any positive integer $k$ , we have the following theorem.

Theorem 3. Let $k\geq 5$ be an integer. For any prime $p$ with $p\equiv 1\bmod\; 2^k$ , if $\chi_{2^k} = \psi$ denotes a $2^k$ -order character modulo $p$ , then we have the identity

$\begin{equation*} \left|\frac{ \mathop{\prod\limits_{j = 1}^{2^{k-1}-1}}\limits_{j\equiv-1\bmod 8}\tau^4\left(\psi^j\right)}{ \mathop{\prod\limits_{h = 1}^{2^{k-1}-1}}\limits_{h\equiv1\bmod 8}\tau^4\left(\psi^h\right)} + \frac{ \mathop{\prod\limits_{j = 1}^{2^{k-1}-1}}\limits_{j\equiv-3\bmod 8}\tau^4\left(\psi^j\right)}{ \mathop{\prod\limits_{h = 1}^{2^{k-1}-1}}\limits_{h\equiv3\bmod 8}\tau^4\left(\psi^h\right)} \right| = \frac{2\cdot |\alpha | }{\sqrt{p}}. \end{equation*}$

2. Several lemmas

In this section, we will give several lemmas by using the relevant properties of character sums.

Lemma 1. Let $p$ be a prime with $p\equiv 1\bmod 4$ , then for any fourth-order character $\chi_4$ modulo $p$ , we have

$\begin{equation*} \tau^2\left(\chi_4\right)+ \tau^2\left(\overline{\chi}_4\right) = 2\sqrt{p}\cdot \alpha, \end{equation*}$

where $\alpha = \frac{1}{2}\sum\limits_{a = 1}^{p-1}\left(\frac{a+\bar{a}}{p}\right)$ , and $\left(\frac{*}{p}\right) = \chi_2$ denotes the Legendre's symbol modulo $p$ .

Proof. See Lemma 3 in [7].

Lemma 2. Let $p$ be a prime with $p\equiv 1 \bmod 4$ . Then for any non-principal character $\psi \bmod p$ , we have the identity

$\begin{equation*} \tau\left(\overline{\psi}^2\right) = \frac{\overline{\psi}(-4)\cdot \sqrt{p}\cdot \tau\left(\overline{\psi}\chi_2\right)}{\tau(\psi)}. \end{equation*}$

Proof. From the properties of the classical Gauss sums we have

$\begin{align} &\sum\limits_{a = 0}^{p-1}\psi\left(a^2-1\right) = \sum\limits_{a = 0}^{p-1}\psi\left((a+1)^2-1\right) = \sum\limits_{a = 1}^{p-1}\psi\left(a^2+2a\right) = \sum\limits_{a = 1}^{p-1}\psi(a)\psi\left(a+2\right)\\ = &\frac{1}{\tau\left(\overline{\psi}\right)}\sum\limits_{a = 1}^{p-1}\psi(a)\sum\limits_{b = 1}^{p-1}\overline{\psi}(b)e\left(\frac{b(a+2)}{p}\right) = \frac{1}{\tau\left(\overline{\psi}\right)}\sum\limits_{b = 1}^{p-1}\overline{\psi}(b)\sum\limits_{a = 1}^{p-1}\psi(a)e\left(\frac{b(a+2)}{p}\right)\\ = &\frac{\tau(\psi)}{\tau\left(\overline{\psi}\right)}\sum\limits_{b = 1}^{p-1}\overline{\psi}(b)\overline{\psi}(b) e\left(\frac{2b}{p}\right) = \frac{\tau(\psi)}{\tau\left(\overline{\psi}\right)}\sum\limits_{b = 1}^{p-1}\overline{\psi}^2(b)e\left(\frac{2b}{p}\right) = \frac{\psi(4)\tau\left(\overline{\psi}^2\right)\tau(\psi)}{\tau\left(\overline{\psi}\right)}. \end{align}$

(2.1)

On the other hand, for any integer $b$ with $(b, p) = 1$ , we have the identity

$\begin{equation*} \sum\limits_{a = 0}^{p-1}e\left(\frac{ba^2}{p}\right) = 1+ \sum\limits_{a = 1}^{p-1}\left(1+\chi_2(a)\right)e\left(\frac{ba}{p}\right) = \sum\limits_{a = 1}^{p-1}\chi_2(a)e\left(\frac{ba}{p}\right) = \chi_2(b)\cdot \sqrt{p}, \end{equation*}$

so note that $\chi_2(-1) = 1$ we also have the identity

$\begin{align} &\sum\limits_{a = 0}^{p-1}\psi\left(a^2-1\right) = \frac{1}{\tau\left(\overline{\psi}\right)}\sum\limits_{a = 0}^{p-1}\sum\limits_{b = 1}^{p-1}\overline{\psi}(b) e\left(\frac{b(a^2-1)}{p}\right)\\ = &\frac{1}{\tau\left(\overline{\psi}\right)}\sum\limits_{b = 1}^{p-1}\overline{\psi}(b)e\left(\frac{-b}{p}\right)\sum\limits_{a = 0}^{p-1} e\left(\frac{ba^2}{p}\right) = \frac{\tau\left(\chi_2\right)}{\tau\left(\overline{\psi}\right)}\sum\limits_{b = 1}^{p-1}\overline{\psi}(b)\chi_2(b)e\left(\frac{-b}{p}\right) \\ = &\frac{\sqrt{p}}{\tau\left(\overline{\psi}\right)}\sum\limits_{b = 1}^{p-1}\overline{\psi}\chi_2(b)e\left(\frac{-b}{p}\right) = \frac{\overline{\psi}(-1)\sqrt{p}\cdot \tau\left(\overline{\psi}\chi_2\right)}{\tau\left(\overline{\psi}\right)}. \end{align}$

(2.2)

Combining (2.1) and (2.2) we have the identity

$\begin{equation*} \tau\left(\overline{\psi}^2\right) = \frac{\overline{\psi}(-4)\cdot \sqrt{p}\cdot \tau\left(\overline{\psi}\chi_2\right)}{\tau(\psi)}. \end{equation*}$

This proves Lemma 2.

Lemma 3. Let $p$ be an odd prime with $p\equiv 1 \bmod 8$ . If $\chi_8$ is an $8$ -order character modulo $p$ , then we have

$\begin{equation*} \left|\tau^4\left(\chi_8\right)+ \tau^4\left(\chi^3_8\right)\right| = 2\cdot p^{\frac{3}{2}}\cdot \left|\alpha\right|. \end{equation*}$

Proof. Taking $\psi = \chi_8$ , then $\psi^2 = \chi_4$ is a four-order character modulo $p$ . Note that $\psi^2(-4) = 1$ , from Lemmas 1 and 2 we have

$\begin{equation*} 2\cdot \sqrt{p}\cdot \alpha = \tau^2\left(\overline{\psi}^2\right)+\tau^2\left(\psi^2\right) = \frac{ p\cdot \tau^2\left(\overline{\psi}\chi_2\right)}{\tau^2(\psi)}+\frac{ p\cdot \tau^2\left(\psi\chi_2\right)}{\tau^2\left(\overline{\psi}\right)} \end{equation*}$

$\begin{equation} \frac{ \tau^2\left(\overline{\psi}\chi_2\right)}{\tau^2(\psi)}+\frac{ \tau^2\left(\psi\chi_2\right)}{\tau^2\left(\overline{\psi}\right)} = \frac{2\cdot\alpha}{\sqrt{p}}. \end{equation}$

(2.3)

Note that $\overline{\psi}\chi_2 = \psi^3$ , $\overline{\psi} = \psi^7$ and $\frac{ \tau^2\left(\chi_8^5\right)}{\tau^2\left(\chi_8^7\right)} = \frac{ \tau^2\left(\chi_8\right)}{\tau^2(\chi^3_8)}$ , from (2.3) we have the identity

$\begin{eqnarray} \tau^4\left(\chi_8\right)+ \tau^4\left(\chi^3_8\right) = \frac{2\cdot\alpha}{\sqrt{p}}\cdot \tau^2\left(\chi_8\right) \tau^2\left(\chi^3_8\right). \end{eqnarray}$

(2.4)

Now Lemma 3 follows from (2.4).

3. Proofs of the theorems

In this section, we shall give the proof of all results by using the properties of the classical Gauss sums, the elementary and analytic methods. For Theorem 1, Let $p$ be an odd prime with $p\equiv 1\bmod 16$ , and $\chi_{16}$ is a sixteen-order character modulo $p$ , then from Lemma 2 we have

$\begin{equation} \tau^4\left(\overline{\chi}_8\right) = \frac{\overline{\chi}_{16}^4(-4)\cdot p^2\cdot \tau^4\left(\chi_{16}^7\right)}{\tau^4\left(\chi_{16}\right)} = \frac{p^2\cdot \tau^4\left(\chi_{16}^7\right)}{\tau^4\left(\chi_{16}\right)} \end{equation}$

(3.1)

and

$\begin{equation} \tau^4\left(\overline{\chi}^3_8\right) = \frac{\overline{\chi}_{16}^{12}(-4)\cdot p^2\cdot \tau^4\left(\chi_{16}^{21}\right)}{\tau^4\left(\chi^3_{16}\right)} = \frac{p^2\cdot \tau^4\left(\chi_{16}^5\right)}{\tau^4\left(\chi^3_{16}\right)}. \end{equation}$

(3.2)

Applying (3.1), (3.2) and Lemma 3 we have

$\begin{equation*} 2\cdot p^{\frac{3}{2}}\cdot \left|\alpha\right| = \left|\tau^4\left(\chi_8\right)+ \tau^4\left(\chi^3_8\right)\right| = \left|\tau^4\left(\overline{\chi}_8\right)+ \tau^4\left(\overline{\chi}^3_8\right)\right| = p^2\cdot \left|\frac{\tau^4\left(\chi_{16}^7\right)}{\tau^4\left(\chi_{16}\right)} + \frac{ \tau^4\left(\chi_{16}^5\right)}{\tau^4\left(\chi^3_{16}\right)}\right| \end{equation*}$

or identity

$\begin{equation*} \left|\frac{\tau^4\left(\chi_{16}^7\right)}{\tau^4\left(\chi_{16}\right)} + \frac{ \tau^4\left(\chi_{16}^5\right)}{\tau^4\left(\chi^3_{16}\right)}\right| = \frac{2\cdot|\alpha|}{\sqrt{p}}. \end{equation*}$

This proves Theorem 1.

Now we prove the Theorem 2. Let $p$ be an odd prime with $p\equiv 1\bmod 32$ , and $\chi_{32}$ is a $32$ -order character modulo $p$ , then from Lemma 2 we have

$\begin{equation} \tau^4\left(\overline{\chi}_{16}\right) = \frac{\overline{\chi}_{32}^4(-4)\cdot p^2\cdot \tau^4\left(\chi_{32}^{15}\right)}{\tau^4\left(\chi_{32}\right)} = \frac{\overline{\chi}_4(2)\cdot p^2\cdot \tau^4\left(\chi_{32}^{15}\right)}{\tau^4\left(\chi_{32}\right)} \end{equation}$

(3.3)

and

$\begin{equation} \tau^4\left(\overline{\chi}^7_{16}\right) = \frac{\overline{\chi}_{32}^{28}(-4)\cdot p^2\cdot \tau^4\left(\chi_{32}^{105}\right)}{\tau^4\left(\chi^7_{32}\right)} = \frac{\chi_4(2)\cdot p^2\cdot \tau^4\left(\chi_{32}^9\right)}{\tau^4\left(\chi^7_{32}\right)}. \end{equation}$

(3.4)

Since $\chi_4^2(2) = \chi_2(2) = 1$ , from (3.3) and (3.4) we get

$\begin{equation} \frac{\tau^4\left(\overline{\chi}^7_{16}\right)}{\tau^4\left(\overline{\chi}_{16}\right)} = \frac{\tau^4\left(\chi_{32}\right)\cdot \tau^4\left(\chi_{32}^9\right)}{\tau^4\left(\chi_{32}^{7}\right)\cdot\tau^4\left(\chi_{32}^{15}\right) } \end{equation}$

(3.5)

and

$\begin{equation} \frac{\tau^4\left(\overline{\chi}^5_{16}\right)}{\tau^4\left(\overline{\chi}^3_{16}\right)} = \frac{\tau^4\left(\chi^3_{32}\right)\cdot \tau^4\left(\chi_{32}^{11}\right)}{\tau^4\left(\chi_{32}^{5}\right)\cdot\tau^4\left(\chi_{32}^{13}\right) }. \end{equation}$

(3.6)

Noting that

$\begin{align} &\left|\frac{\tau^4\left(\overline{\chi}_{32}\right)\cdot \tau^4\left(\overline{\chi}_{32}^9\right)}{\tau^4\left(\overline{\chi}_{32}^{7}\right)\cdot\tau^4\left(\overline{\chi}_{32}^{15}\right) }+ \frac{\tau^4\left(\overline{\chi}^3_{32}\right)\cdot \tau^4\left(\overline{\chi}_{32}^{11}\right)}{\tau^4\left(\overline{\chi}_{32}^{5}\right)\cdot\tau^4\left(\overline{\chi}_{32}^{13}\right) } \right|\\ = &\left|\frac{\tau^4\left(\chi^7_{32}\right)\cdot \tau^4\left(\chi_{32}^{15}\right)}{\tau^4\left(\chi_{32}\right)\cdot\tau^4\left(\chi_{32}^{9}\right) }+ \frac{\tau^4\left(\chi^{5}_{32}\right)\cdot \tau^4\left(\chi_{32}^{13}\right)}{\tau^4\left(\chi_{32}^{3}\right)\cdot\tau^4\left(\chi_{32}^{11}\right) } \right|. \end{align}$

(3.7)

Combining (3.5)–(3.7) and Theorem 1 we immediately have the identity

$\begin{align*} &\left|\frac{\tau^4\left(\chi^7_{32}\right)\cdot \tau^4\left(\chi_{32}^{15}\right)}{\tau^4\left(\chi_{32}\right)\cdot\tau^4\left(\chi_{32}^{9}\right) }+ \frac{\tau^4\left(\chi^{5}_{32}\right)\cdot \tau^4\left(\chi_{32}^{13}\right)}{\tau^4\left(\chi_{32}^{3}\right)\cdot\tau^4\left(\chi_{32}^{11}\right) } \right|\\ = &\left|\frac{\tau^4\left(\overline{\chi}_{32}\right)\cdot \tau^4\left(\overline{\chi}_{32}^9\right)}{\tau^4\left(\overline{\chi}_{32}^{7}\right)\cdot\tau^4\left(\overline{\chi}_{32}^{15}\right) }+ \frac{\tau^4\left(\overline{\chi}^3_{32}\right)\cdot \tau^4\left(\overline{\chi}_{32}^{11}\right)}{\tau^4\left(\overline{\chi}_{32}^{5}\right)\cdot\tau^4\left(\overline{\chi}_{32}^{13}\right) } \right|\\ = &\left|\frac{\tau^4\left(\chi_{32}\right)\cdot \tau^4\left(\chi_{32}^9\right)}{\tau^4\left(\chi_{32}^{7}\right)\cdot\tau^4\left(\chi_{32}^{15}\right) }+ \frac{\tau^4\left(\chi^3_{32}\right)\cdot \tau^4\left(\chi_{32}^{11}\right)}{\tau^4\left(\chi_{32}^{5}\right)\cdot\tau^4\left(\chi_{32}^{13}\right) } \right| \\ = &\left|\frac{\tau^4\left(\overline{\chi}_{16}^7\right)}{\tau^4\left(\overline{\chi}_{16}\right)} + \frac{ \tau^4\left(\overline{\chi}_{16}^5\right)}{\tau^4\left(\overline{\chi}^3_{16}\right)}\right| = \left|\frac{\tau^4\left(\chi_{16}^7\right)}{\tau^4\left(\chi_{16}\right)} + \frac{ \tau^4\left(\chi_{16}^5\right)}{\tau^4\left(\chi^3_{16}\right)}\right| = \frac{2\cdot|\alpha|}{\sqrt{p}}. \end{align*}$

This proves Theorem 2.

From Theorems 1 and 2 we know that Theorem 3 is true for $k = 4$ and $5$ . Assuming that Theorem 3 holds true for $k = n\geq 5$ . That is,

$\begin{equation} \left|\frac{ \mathop{\prod\limits_{j = 1}^{2^{n-1}-1}}\limits_{j\equiv-1\bmod 8}\tau^4\left(\psi^j\right)}{ \mathop{\prod\limits_{h = 1}^{2^{n-1}-1}}\limits_{h\equiv1\bmod 8}\tau^4\left(\psi^h\right)} + \frac{ \mathop{\prod\limits_{j = 1}^{2^{n-1}-1}}\limits_{j\equiv-3\bmod 8}\tau^4\left(\psi^j\right)}{ \mathop{\prod\limits_{h = 1}^{2^{n-1}-1}}\limits_{h\equiv3\bmod 8}\tau^4\left(\psi^h\right)} \right| = \frac{2\cdot |\alpha | }{\sqrt{p}}, \end{equation}$

(3.8)

where $\psi = \chi_{2^n}$ is a $2^n$ -order character modulo $p$ .

So the conjugate of (3.8) is

$\begin{equation} \left|\frac{ \mathop{\prod\limits_{j = 1}^{2^{n-1}-1}}\limits_{j\equiv-1\bmod 8}\tau^4\left(\overline{\psi}^j\right)}{ \mathop{\prod\limits_{h = 1}^{2^{n-1}-1}}\limits_{h\equiv1\bmod 8}\tau^4\left(\overline{\psi}^h\right)} + \frac{ \mathop{\prod\limits_{j = 1}^{2^{n-1}-1}}\limits_{j\equiv-3\bmod 8}\tau^4\left(\overline{\psi}^j\right)}{ \mathop{\prod\limits_{h = 1}^{2^{n-1}-1}}\limits_{h\equiv3\bmod 8}\tau^4\left(\overline{\psi}^h\right)} \right| = \frac{2\cdot |\alpha | }{\sqrt{p}}. \end{equation}$

(3.9)

Then for $k = n+1$ and $j$ with $j\equiv 1\bmod 8$ , from Lemma 2 we have

$\begin{equation} \frac{\tau^4\left(\overline{\chi}^{7j}_{2^n}\right)}{\tau^4\left(\overline{\chi}^j_{2^n}\right)} = \frac{\overline{\chi}^{3j}_{2^{n-1}}\left(2^4\right)\cdot \tau^4\left(\chi^j_{2^{n+1}}\right)\cdot \tau^4\left(\chi^{j(2^n-2^3+1)}_{2^{n+1}}\right)}{\tau^4\left(\chi^{7j}_{2^{n+1}}\right)\cdot \tau^4\left(\chi^{j(2^n-1)}_{2^{n+1}}\right)}. \end{equation}$

(3.10)

Note that the identity

$\prod\limits_{s = 0}^{2^{n-3}-1}\overline{\chi}^{3(8s+1)}_{2^n}\left(2^4\right) = \overline{\chi}^{2^{n-3}+2^{n-1}\left(2^{n-1}-1\right)}_{2^n}\left(2^{12}\right) = 1.$

From (3.9) and (3.10) we may immediately deduce the identity

$\begin{equation} \left|\frac{ \mathop{\prod\limits_{j = 1}^{2^{n}-1}}\limits_{j\equiv-1\bmod 8}\tau^4\left(\overline{\chi}_{2^{n+1}}^j\right)}{ \mathop{\prod\limits_{h = 1}^{2^{n}-1}}\limits_{h\equiv1\bmod 8}\tau^4\left(\overline{\chi}_{2^{n+1}}^h\right)} + \frac{ \mathop{\prod\limits_{j = 1}^{2^{n}-1}}\limits_{j\equiv-3\bmod 8}\tau^4\left(\overline{\chi}_{2^{n+1}}^j\right)}{ \mathop{\prod\limits_{h = 1}^{2^{n}-1}}\limits_{h\equiv3\bmod 8}\tau^4\left(\overline{\chi}_{2^{n+1}}^h\right)} \right| = \frac{2\cdot |\alpha | }{\sqrt{p}}. \end{equation}$

(3.11)

Taking the conjugate for (3.11), this implies that Theorem 3 is complete for $k = n+1$ .

This proves Theorem 3 by mathematical induction.

4. Conclusions

The main result of this paper is to prove a new identity for the classical Gauss sums. That is, if $p$ is a prime with $p\equiv 1\bmod 2^k$ , $k\geq 4$ , then for any $2^k$ -order character $\psi$ modulo $p$ , we have the identity

These results give the exact values of some special Gauss sums and reveal the values distribution of classical Gauss sums. This not only promotes the research of some well-known sums, such as Kloosterman sums, which plays an important role in the study of the Diophantine equation, but also makes new contributions to the research of other related fields.

Acknowledgments

The author would like to express her gratitude to the referees and coordinating editor for their valuable and detailed comments. This work was supported by the National Natural Science Foundation of China under Grant No.11771351.

Conflict of interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author did not receive support from any organization for the submitted work.

Conflict of interest

Khue Vu Nguyen is an editorial board member for AIMS Neuroscience and was not involved in the editorial review or the decision to publish this article. The author declares that there are no competing interests.

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Epigenetic modulation of human neurobiological disorders: Lesch-Nyhan disease as a model disorder

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Epigenetic modulation of human neurobiological disorders: Lesch-Nyhan disease as a model disorder

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