Open Access
 Issue Wuhan Univ. J. Nat. Sci. Volume 27, Number 1, March 2022 49 - 52 https://doi.org/10.1051/wujns/2022271049 16 March 2022

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## 0 Introduction

Let p be an odd prime. Let be a finite field of q elements with and denote the set of all the nonzero elements of . Let N(f = b) denote the number of solutions of the equation in where is a polynomial in and . That is,Studying the value of N(f = b) is one of the main topics in finite fields. Generally speaking, it is nontrivial to give the formula for N(f = b). Finding the explicit formula for N(f = b) under certain condition has attracted lots of authors for many years.

Markoff-Hurwitz-type equations belong to the following type of the Diophantine equationswhere n, a are positive integers and . This type of equations was first studied by Markoff[1] for the case . More generally, these equations were studied by Hurwitz[2].

Recently, Baoulina[3-5] studied the generalized Markoff-Hurwitz-type equationswhere and are positive integers satisfying for and . Baoulina[4-6] and Pan et al[7] considered the further generalized Markoff- Hurwitz-type equations of the form:(1)where are positive integers, , for . The special case (1) of is investigated by Cao[8]. Song and Chen[9] presented the formulas for the number of solutions of the following equationsover the finite field under some certain restrictions, where for and

Hu and Li[10] consider the rational points of the further generalized Markoff-Hurwitz-type equations of the formover the finite field under some certain cases, where , and are positive integers, , for

In this paper, we consider the number of solutions of the following equations(2)over the finite field under some other restrictions, where , , are positive integers. In what follows, we always let

For any positive integers we let denote the number of of integers with such that is an integer. Denote by the number of solutions of (2) in . Our main result is the following theorem.

Theorem 1  　Suppose that and there is a positive integer l such that , with l as chosen minimal. Then and

This paper is organized as follows. In Section 1, we review some useful known lemmas which will be needed later. Subsequently, in Section 2, we prove Theorem 1. Some interesting applications of Theorem 1 will be provided as corollaries at the end of this paper.

## 1 Preliminary Lemmas

In this section, we present some useful lemmas that are needed in the proof of Theorem 1 as follows.

Lemma 1  [9,11]　For any positive integer m, the number of elements of m-th power in is .

Lemma 2  [10]　Let be positive integers and . Then for any elements we have

The following two lemmas are the main results in Ref.[6] and fundamental for our results.

Lemma 3  [6]　Let Suppose that there is a positive integer l such that and . Then

Lemma 4  [6]　Suppose that and there is a positive integer l such that , with l chosen minimal. Then andwhereand

## 2 Proof of Theorem 1

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

Proof of Theorem 1  　Let (resp. ) denote the number of the solutions of the equations with (resp. ). Clearly, one has(3)

Then we can solve the problem in two cases. One is and the other one is .

Case (i) . Then one has(4)

Using the assumption there is a positive integer l such that and . Thus, by (4) and Lemma 3 ,(5)

Case (ii) If , we let . Define

Note that Then from Lemma 2, we can deduce that(6)

Noting that integer l such that , with l chosen minimal. Thus for any given from Lemma 1 and Lemma 3, one has(7)

Then by (6) together with (7), we have(8)

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

To conclude this section, we present some corollaries. It is clear that plays a central role in Theorem 1. For any positive integers Sun and Wan[12] showed that if and only if either for some j or t is odd, are pairwise coprime, and each is coprime with any odd number in , where is the set of even integers among . In Ref.[13], they also showed that if and only if are pairwise coprime, and at least (r-1) of the are odd.

Therefore we can easily deduce the following corollary.

Corollary 1  　Suppose that are odd, are even, are pairwise coprime, . Under the conditions of Theorem 1, we havewhereand

Let v be a positive integer. It is also known (Ref. [14], Proposition 6.17) that

Then we have the second corollary.

Corollary 2  　Suppose that Under the conditions of Theorem 1, we have

Clearly, Corollaries 1-2 are some special cases of Theorem 1. For example, consider the further generalized Markoff-Hurwitz-type equation over (9)

Clearly Then we get (5, 2, 6)=30, and One can immediately conclude that (9) has 6 817 solutions in by Corollary 1.

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