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  浙江大学学报(理学版)  2017, Vol. 44 Issue (5): 516-519, 537  DOI:10.3785/j.issn.1008-9497.2017.05.003
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HU Shuangnian, LI Yanyan. The number of solutions of generalized Markoff-Hurwitz-type equations over finite fields[J]. Journal of Zhejiang University(Science Edition), 2017, 44(5): 516-519, 537. DOI: 10.3785/j.issn.1008-9497.2017.05.003.
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胡双年, 李艳艳. 有限域上广义Markoff-Hurwitz-type方程的有理点个数[J]. 浙江大学学报(理学版), 2017, 44(5): 516-519, 537. DOI: 10.3785/j.issn.1008-9497.2017.05.003.
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Fundation item

Supported by the Key Program of Universities of Henan Province of China (17A110010), China Postdoctoral Science Foundation Funded Project (2016M602251) and by the National Science Foundation of China Grant (11501387)

About the author

HU Shuangnian(1982-), ORCID:http://orcid.org/0000-0002-5174-8460, male, Ph.D, lecturer, the field of interest is number theory, E-mail:hushuangnian@163.com

Article History

Received Date: November 7, 2016
The number of solutions of generalized Markoff-Hurwitz-type equations over finite fields
HU Shuangnian1,2 , LI Yanyan3     
1. School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004, Henan Province, China;
2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China;
3. School of Electronic and Electrical Engineering, Nanyang Institute of Technology, Nanyang 473004, Henan Province, China
Received Date: November 7, 2016
Fundation item: Supported by the Key Program of Universities of Henan Province of China (17A110010), China Postdoctoral Science Foundation Funded Project (2016M602251) and by the National Science Foundation of China Grant (11501387)
About the author: HU Shuangnian(1982-), ORCID:http://orcid.org/0000-0002-5174-8460, male, Ph.D, lecturer, the field of interest is number theory, E-mail:hushuangnian@163.com
Abstract: Let Nq denote the number of solutions of the generalized Markoff-Hurwitz-type equations (a1x1m1+a2x2m2+…+anxnmn)k=cx1k1 x2k2xtkt over the finite field $\mathbb{F}$q, where n ≥ 2, mi, k, kj and tn are positive integers, ai, c$\mathbb{F}$q*, for 1 ≤ in and 1 ≤ jt.Recently, some researches considered the above equation with k=k1=…=kt=1 and obtained some generalizations of Carlitz's results.In this paper, we determine Nq explicitly in some other cases.This extends the previous conclusions.
Key words: finite field    rational point    Markoff-Hurwitz-type equations    
有限域上广义Markoff-Hurwitz-type方程的有理点个数
胡双年1,2, 李艳艳3    
1. 南阳理工学院 数学与统计学院, 河南 南阳 473004;
2. 郑州大学 数学与统计学院, 河南 郑州 450001;
3. 南阳理工学院 电子与电气工程学院, 河南 南阳 473004
摘要: 设Nq表示有限域$\mathbb{F}$q上广义Markoff-Hurwitz-type方程的有理点个数(a1x1m1+a2x2m2+…+anxnmnk=cx1k1 x2k2xtkt,其中n ≥ 2,mikkjtn是正整数,aic属于$\mathbb{F}$q*,其中1 ≤ in,1 ≤ jt.最近有研究推广了Carlitz的结果,给出了上述方程当k=k1=…=kt=1时的有理点个数.当未定元的指数满足一定条件时,本文给出了上述广义方程的有理点个数,推广了已有结论.
关键词: 有限域    有理点    Markoff-Hurwitz-type方程    
0 Introduction and the main result

Markoff-Hurwitz-type equations are the following type of the Diophantine equation:

$ x_1^2 + x_2^2 + \cdots + x_n^2 = c{x_1}{x_2} \cdots {x_n}, $

where n, c are positive integers and n≥3. This type of equations were firstly studied by MARKOFF[1] for the case n=3, c=3 because of its relation to Diophantine approximation. More generally, these equations were studied by HURWITZ [2].

CARLITZ[3] investigated the Markoff-Hurwitz-type equations over finite fields. Let Nq denote the finite of q elements and $\mathbb{F}$q*=$\mathbb{F}$q{0}, He investigated the number of $\mathbb{F}$q-rational points on the equation

$ {a_1}x_1^2 + {a_2}x_2^2 + \cdots + {a_n}x_n^2 = c{x_1}{x_2} \cdots {x_n}, $

where a1, a2, …, an, c$\mathbb{F}$q* and n≥3, and obtained the formulas in terms of Jacobsthal sum for n=3 and n=4. CARLITZ[4] studied equations of the form (x1+x2+…+xn)2=bx1x2xn over finite fields of odd characteristic.

Recently, BAOULINA[5-7] studied the generalized Markoff-Hurwitz-type equation:

$ {a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}} = c{x_1}{x_2} \cdots {x_n}, $

where ai, c$\mathbb{F}$q*, mi are positive integers which satisfied mi|(q-1) for i=1, 2, …, n and n≥2. PAN et al[8] considered the further generalized Markoff-Hurwitz-type equations of the form:

$ {\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)^k} = cx_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}, $ (1)

where n≥2, mi, ki, k are positive integers, ai, c$\mathbb{F}$q*, for 1≤in. The special case (1) of k=1 is investigated by CAO[9]. For positive integers d1, d2, …, dk, let I(d1, d2, …, dk) denote the number of k-tuples (j1, j2, …, jk) of integers with 1≤jrdr-1(1≤rk) such that $\left( \frac{{{j}_{1}}}{{{d}_{1}}} \right)+\left( \frac{{{j}_{2}}}{{{d}_{2}}} \right)+\cdots +\left( \frac{{{j}_{k}}}{{{d}_{k}}} \right)$ is an integer. Using the function I(mj1, mj2, …, mjr)(1≤j1 < … < jrn), SONG et al[10] presented the formulas for the number of solutions of the following equation defined over $\mathbb{F}$q:

$ x_1^{{m_1}} + x_2^{{m_2}} + \cdots + x_n^{{m_n}} = c{x_1}{x_2} \cdots {x_t} $

under some certain restrictions, where mj|(q-1), n≥2, c$\mathbb{F}$q*, t>n. Some other generalizations of Markoff-Hurwitz-type equations were reported in [11-12].

In this paper, we consider the rational points of the further generalized Markoff-Hurwitz-type equations of the form

$ {\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)^k} = cx_1^{{k_1}}x_2^{{k_2}} \cdots x_t^{{k_t}} $ (2)

over the finite field $\mathbb{F}$q in some other cases, where n≥2, mi, kj, k, and tn, are positive integers, ai, c$\mathbb{F}$q*, for 1≤in, 1≤jt.

Let f(x1, x2, …, xn) and g(x1, x2, …, xn) be polynomials with coefficients in $\mathbb{F}$q. Throughout this paper, we use Nq[f(x1, x2, …, xn)=g(x1, x2, …, xn)] to denote the number of the rational points of the equation f(x1, x2, …, xn)=g(x1, x2, …, xn) in $\mathbb{F}$qmax(n, s). We also let di=gcd(mi, q-1), M=lcm[m1, m2, …, mn] and Nq denote the number of solutions in $\mathbb{F}$qt to (2) as above. Using the results of ref. [8], we will show that the expression of Nq becomes very simple under some restrictions on the exponents. Our main result is

Theorem 1   With the notation as above, if $\gcd \left( \sum\limits_{i=1}^{n}{\frac{{{k}_{i}}M}{{{m}_{i}}}-kM, q-1} \right)=1$ and d1, d2, …, dn are pairwise coprime, then we have

$ {N_q} = {q^{t - 1}} + {\left( { - 1} \right)^{n - 1}}{\left( {q - 1} \right)^{t - n}}. $

Clearly, Nq is independent of the coefficients ai, c and the exponents kn+1, …, kt. Letting t=n, then theorem 1 reduces to the theorem of PAN et al[8]. Theorem 1 also generalizes the main results of [10] in some other cases.

This paper is organized as follows. In section 1, we recall some useful known lemmas. In section 2, we make use of the results presented in section 1 to show theorem 1. Some interesting applications of theorem 1 will be provided as corollaries at the end.

1 Preliminary lemmas

In this section, we present some useful lemmas which are needed in section 2. Let m be a positive integer and h(x1, x2, …, xr) be a polynomial with integer coefficients. We use N[h≡0(mod m)] to denote the number of the solutions of the congruence h(x1, x2, …, xr)≡(mod m). We first recall two well known results in the elementary number theory.

Lemma 1[13]   Let a, b be positive integers. Then

$ \gcd \left( {a,b} \right){\rm{lcm}}\left[ {a,b} \right] = ab. $

Lemma 2[13]   Let a1, a2, …, ar, m, b be positive integers. Then the necessary and sufficient condition for the congruence a1x1+a2x2+…+arxrb(mod m) to have a solution is that d=gcd(a1, a2, …, ar, m)|b. If this condition is satisfied, then the number of incongruent (mod m) solutions is mr-1d. Furthermore, we have

$ N\left[ {\sum\limits_{i = 1}^r {{a_i}{x_i}} \equiv b\left( {\bmod m} \right)} \right] = N\left[ {d\sum\limits_{i = 1}^r {{x_i} \equiv b\left( {\bmod m} \right)} } \right]. $

Lemma 3   Let t1, t2, …, tr be positive integers and d=gcd(t1, t2, …, tr, q-1). Then for any elements a, α$\mathbb{F}$q*, we have

$ \begin{gathered} {N_q}\left[ {ax_1^{{t_1}}x_2^{{t_2}} \cdots x_r^{{t_r}} = \alpha } \right] = {N_q}\left[ {a{{\left( {{x_1}{x_2} \cdots {x_r}} \right)}^d} = \alpha } \right] = \hfill \\ \left\{ \begin{gathered} d{\left( {q - 1} \right)^{r - 1}},\;\;{\text{if}}\;{a^{ - 1}}\alpha \;{\text{is}}\;{\text{a}}\;d - {\text{th}}\;{\text{power}}\;{\text{in}}\;\mathbb{F}_q^ * , \hfill \\ 0,\;\;{\text{otherwise}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $

Proof   Using the fact that $\mathbb{F}$q* is cyclic, we can let ξ be a primitive element of $\mathbb{F}$q*. Then for any element b$\mathbb{F}$q*, we have b=ξk, where k is a positive integer such that 1≤kq-1. Suppose that a-1α=ξm, then using lemma 2, we have

$ \begin{array}{l} {N_q}\left[ {ax_1^{{t_1}} \cdots x_r^{{t_r}} = \alpha } \right] = \\ \sum\limits_{1 \le {e_1}, \cdots ,{e_r} \le q - 1} {{N_q}\left[ {{\xi ^{{e_1}{t_1}}} \cdots {\xi ^{{e_r}{t_r}}} = {\xi ^m}} \right]} = \\ \sum\limits_{1 \le {e_1}, \cdots ,{e_r} \le q - 1} {{N_q}\left[ {{e_1}{t_1} + \cdots + {e_r}{t_r} = m\left( {\bmod q - 1} \right)} \right]} = \\ \sum\limits_{1 \le {e_1}, \cdots ,{e_r} \le q - 1} {{N_q}\left[ {\left( {{e_1} + \cdots + {e_r}} \right)d = m\left( {\bmod q - 1} \right)} \right]} = \\ \sum\limits_{1 \le {e_1}, \cdots ,{e_r} \le q - 1} {{N_q}\left[ {{{\left( {{\xi ^{{e_1}}} \cdots {\xi ^{{e_r}}}} \right)}^d} = {\xi ^m}} \right]} = \\ {N_q}\left[ {a{{\left( {{x_1}{x_2} \cdots {x_r}} \right)}^d} = \alpha } \right]. \end{array} $

Since lemma 2 tells us that

$ \begin{array}{l} \sum\limits_{1 \le {e_1}, \cdots ,{e_r} \le q - 1} {{N_q}\left[ {{e_1}{t_1} + \cdots + {e_r}{t_r} = m\left( {\bmod q - 1} \right)} \right]} = \\ \;\;\;\;\;\;\left\{ \begin{array}{l} d{\left( {q - 1} \right)^{r - 1}},\;\;{\rm{if}}\;d\left| m \right.,\\ 0,\;\;{\rm{otherwise}}. \end{array} \right. \end{array} $

Then the desired result follows immediately. This ends the proof of lemma 2.

Lemma 4[14]   Let n be a positive integer and 1≤jn. Assume that $\gcd \left( {{d}_{j}}, \frac{{{d}_{1}}\cdots {{d}_{n}}}{{{d}_{j}}} \right)=1$ for some j. Then

$ \begin{array}{l} {N_q}\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}} = 0} \right) = \\ \;\;\;\;\;\;\;\;{N_q}\left( {{a_1}x_1^{{d_1}} + {a_2}x_2^{{d_2}} + \cdots + {a_n}x_n^{{d_n}} = 0} \right) = {q^{n - 1}}. \end{array} $

The following result comes from PAN et al [8].

Lemma 5[8]   In the equation (2), if t=n and $\gcd \left( \sum\limits_{i=1}^{n}{\frac{{{m}_{1}}\cdots {{m}_{n}}}{{{m}_{i}}}-k{{m}_{1}}\cdots {{m}_{n}}, q-1} \right)=1$, then we have

$ \begin{array}{l} {N_q}\left[ {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}} = cx_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;{q^{n - 1}} + {\left( { - 1} \right)^{n - 1}}. \end{array} $

Lemma 6[10, 15]   The number of elements of d-th power in $\mathbb{F}$q* is $\frac{\left( q-1 \right)}{d}$.

2 Proof of theorem 1

In this section, we give the proof of theorem 1.

Proof   Firstly, we claim that the condition of lemma 5 is equivalent to the conditions of theorem 1. That is, the condition

$ \gcd \left( {\sum\limits_{i = 1}^n {\frac{{{k_i}{m_1} \cdots {m_n}}}{{{m_i}}}} - k{m_1} \cdots {m_n},q - 1} \right) = 1 $

is equivalent to the following two conditions:

$ \gcd \left( {\sum\limits_{i = 1}^n {\frac{{{k_i}M}}{{{m_i}}}} - kM,q - 1} \right) = 1\;{\rm{and}}\;{d_1},{d_2}, \cdots ,{d_n} $

are pairwise coprime. Since

$ \sum\limits_{i = 1}^n {\frac{{{k_i}{m_1} \cdots {m_n}}}{{{m_i}}}} - k{m_1} \cdots {m_n} = \frac{{{m_1} \cdots {m_n}}}{M}\left( {\sum\limits_{i = 1}^n {\frac{{{k_i}M}}{{{m_i}}}} - kM} \right), $

thus we can deduce that the condition

$ \gcd \left( {\sum\limits_{i = 1}^n {\frac{{{k_i}{m_1} \cdots {m_n}}}{{{m_i}}}} - k{m_1} \cdots {m_n},q - 1} \right) = 1 $

is equivalent to $\gcd \left( \sum\limits_{i=1}^{n}{\frac{{{k}_{i}}M}{{{m}_{i}}}-kM, q-1} \right)=1$ and $\gcd \left( \frac{{{m}_{1}}\cdots {{m}_{n}}}{M}, q-1 \right)=1$. From lemma 1, one can deduce that M·gcd(mi, mj)|m1mn. Since gcd(di, dj)|(q-1) and gcd(di, dj)|gcd (mi, mj), thus one can easily deduce that $\gcd \left( \frac{{{m}_{1}}\cdots {{m}_{n}}}{M}, q-1 \right)=1$ if and only if the integers d1, d2, …, dn are pairwise coprime. This completes the proof of the claim.

In the following, we use Nq (resp. Ñq) to denote the number of the solutions of the equations (a1x1m1+a2x2m2+…+anxnmn)k=cx1k1x2k2xtkt with xkn+1n+1xtkt=0 (resp. xkn+1n+1xtkt≠0). Clearly, one has

$ {N_q} = {{\bar N}_q} + {{\tilde N}_q}. $ (3)

Then, we can solve the problem in two cases. One is xkn+1n+1xtkt=0 and the other one is xkn+1n+1xtkt≠0.

Case ⅰ   Clearly, xkn+1n+1xtkt=0 is reduced to say that xn+1xt=0. Then, we have

$ \begin{array}{l} {N_q}\left[ {{x_{n + 1}} \cdots {x_t} = 0} \right] = \sum\limits_{j = 1}^{t - n} {\left( {\begin{array}{*{20}{c}} {t - n}\\ j \end{array}} \right){{\left( {q - 1} \right)}^{t - n - j}}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{q^{t - n}} - {\left( {q - 1} \right)^{t - n}}. \end{array} $ (4)

Using the assumption d1, d2, …, dn are pairwise coprime, it follows from (4) and lemma 4 that

$ \begin{array}{l} {{\bar N}_q} = \left( {{q^{t - n}} - {{\left( {q - 1} \right)}^{t - n}}} \right) \times \\ \;\;\;\;\;\;\;\;{N_q}\left[ {{{\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)}^k} = 0} \right] = \\ \;\;\;\;\;\;\;\;\left( {{q^{t - n}} - {{\left( {q - 1} \right)}^{t - n}}} \right) \times \\ \;\;\;\;\;\;\;\;{N_q}\left[ {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}} = 0} \right] = \\ \;\;\;\;\;\;\;\;{q^{n - 1}}\left( {{q^{t - n}} - {{\left( {q - 1} \right)}^{t - n}}} \right) = \\ \;\;\;\;\;\;\;\;{q^{t - 1}} - {q^{n - 1}}{\left( {q - 1} \right)^{t - n}}. \end{array} $ (5)

Case ⅱ If xkn+1n+1xtkt≠0, we can let f=cxkn+1n+1xtkt and d=gcd(kn+1, …, kt, q-1).

Define S:={a$\mathbb{F}$q*:ac-1 be a d-th power element in $\mathbb{F}$q*}. From lemma 3, we can deduce that

$ \begin{array}{l} {{\tilde N}_q} = {N_q}\left[ {{{\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)}^k} = fx_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}} \right] = \\ \;\;\;\;\;\;\;\;\;d{\left( {q - 1} \right)^{t - n - 1}} \times \\ \;\;\;\;\;\;\;\;\;\sum\limits_{a \in S} {{N_q}\left[ {{{\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)}^k} = } \right.} \\ \;\;\;\;\;\;\;\;\;\left. {ax_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}} \right]. \end{array} $ (6)

Noting that $\gcd \left( \sum\limits_{i=1}^{n}{\frac{{{k}_{i}}{{m}_{1}}\cdots {{m}_{n}}}{{{m}_{i}}}-kM, q-1} \right)=1$ and d1, d2, …, dn are pairwise coprime. Thus, for any given aS, from the claim and lemma 5, one has

$ \begin{array}{l} {N_q}\left[ {{{\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)}^k} = } \right.\\ \;\;\;\;\;\;\;\left. {ax_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}} \right] = {q^{n - 1}} + {\left( { - 1} \right)^{n - 1}}. \end{array} $

Then, it follows from lemma 6 that

$ \begin{array}{l} \sum\limits_{a \in S} {{N_q}\left[ {{{\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)}^k} = } \right.} \\ \;\;\;\;\;\;\;\left. {cx_1^{{k_1}}x_2^{{k_2}} \cdots x_n^{{k_n}}} \right] = \frac{{q - 1}}{d}\left( {{q^{n - 1}} + {{\left( { - 1} \right)}^{n - 1}}} \right). \end{array} $ (7)

Using (6) and (7), one can derive that

$ {{\tilde N}_q} = {\left( {q - 1} \right)^{t - n}}\left( {{q^{n - 1}} + {{\left( { - 1} \right)}^{n - 1}}} \right). $ (8)

The desired result can follow immediately from (3), (5) and (8). This ends the proof of theorem 1.

In concluding this section, we present some trivial corollaries.

Corollary 1   For the further generalized Markoff-Hurwitz-type equations of the form

$ {\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)^k} = c{x_1}{x_2} \cdots {x_t} $

over the finite field $\mathbb{F}$q, where n≥2, mi, k and tn are positive integers, ai, c$\mathbb{F}$q*, for 1≤in. If $\gcd \left( \sum\limits_{i=1}^{n}{\frac{M}{{{m}_{i}}}-kM, q-1} \right)=1$ and d1, d2, …, dn are pairwise coprime, then

$ {N_q} = {q^{t - 1}} + {\left( { - 1} \right)^{n - 1}}{\left( {q - 1} \right)^{t - n}}. $

Corollary 2   For the further generalized Markoff-Hurwitz-type equations of the form

$ {{\left( {{a}_{1}}{{x}_{1}}^{{{m}_{1}}}+{{a}_{2}}{{x}_{2}}^{{{m}_{2}}}+\cdots +{{a}_{n}}{{x}_{n}}^{{{m}_{n}}} \right)}^{k}}=c{{x}_{1}}^{{{m}_{1}}}{{x}_{2}}^{{{m}_{2}}}\cdots {{x}_{t}}^{{{m}_{t}}} $

over the finite field $\mathbb{F}$q, where n≥2, mi, k and tn are positive integers, ai, c$\mathbb{F}$q* for 1≤in. If gcd((n-k)M, q-1)=1 and d1, d2, …, dn are pairwise coprime, then

$ {N_q} = {q^{t - 1}} + {\left( { - 1} \right)^{n - 1}}{\left( {q - 1} \right)^{t - n}}. $

Corollary 3   For the further generalized Markoff-Hurwitz-type equations of the form

$ {\left( {{a_1}x_1^{{m_1}} + {a_2}x_2^{{m_2}} + \cdots + {a_n}x_n^{{m_n}}} \right)^k} = c{x_1}{x_2} \cdots {x_t} $

over the finite field $\mathbb{F}$q, where n≥2, mi, k and tn are positive integers, ai, c$\mathbb{F}$q*, for 1≤in. If gcd((n-km)m, q-1)=1, then

$ {N_q} = {q^{t - 1}} + {\left( { - 1} \right)^{n - 1}}{\left( {q - 1} \right)^{t - n}}. $

Clearly, corollaries 1~3 are some special cases of theorem 1. For example, consider the further generalized Markoff-Hurwitz-type equation over

$ {\mathbb{F}_{13}}:\left( {{a_1}x_1^5 + {a_2}x_2^3 + {a_3}x_3^8} \right) = bx_1^7x_2^7x_3^7x_4^{10}x_5^5x_6^4, $ (8)

where a1, a2, a3, b$\mathbb{F}$13*. Since gcd(5, 12)=1, gcd(3, 12)=3, gcd(8, 12)=4 and gcd(7(24+40+15)-240, 12)=1 are co-prime, one can immediately conclude that (8) has 373021 solutions in $\mathbb{F}$136 by theorem 1.

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