Abstract
Let s be a positive integer, p be an odd prime, [see formula in PDF], and let [see formula in PDF] be a finite field of q elements. Let [see formula in PDF] be the number of solutions of the following equations: [see formula in PDF][see formula in PDF] over the finite field [see formula in PDF], with [see formula in PDF][see formula in PDF][see formula in PDF] and [see formula in PDF][see formula in PDF] are positive integers. In this paper, we find formulas for [see formula in PDF] when there is a positive integer l such that dD|([see formula in PDF]) , where [see formula in PDF][see formula in PDF][see formula in PDF][see formula in PDF][see formula in PDF][see formula in PDF][see formula in PDF][see formula in PDF] And we determine [see formula in PDF] explicitly under certain cases. This extends Markoff-Hurwitz-type equations over finite field.
Highlights
We find formulas for Nq when there is a positive integer l such that dD|( pl 1 ), where D lcm [d1, dn ]
Let Fq be a finite field of q elements with q ps, s ≥1 and Fq* denote the set of all the nonzero elements of Fq
Markoff-Hurwitz-type equations belong to the following type of the Diophantine equations x12 x22 xn2 ax1x2 xn where n, a are positive integers and n ≥ 3
Summary
Hu and Li[10] consider the rational points of the further generalized Markoff-Hurwitz-type equations of the form x kt t over the finite field Fq under some certain cases, where n ≥ 2 , mi , k j , k and t n are positive integers, ai , a Fq*, for 1≤ i ≤ n, 1≤ j ≤ t. Over the finite field Fq under some other restrictions, where n ≥ 2 , t n , k, k j (n 1≤ j ≤ t), mi (1≤ i ≤ n) are positive integers. Theorem 1 Suppose that gcd(kn 1, ,kt ) d , dD 2 and there is a positive integer l such that dD | ( pl 1) , with l as chosen minimal. Some interesting applications of Theorem 1 will be provided as corollaries at the end of this paper
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