© Hindawi Publishing Corp. ON THE SPECIAL SOLUTIONS OF AN EQUATION IN A FINITE FIELD

The main purpose of this paper is to prove the following 
conclusion: let p be a prime large enough and let k be a fixed positive integer with 2 k | p − 1 . Then for any finite field F p and any element 0 ≠ c ∈ F p , there exist three generators x , y , and z ∈ F p such that x k y k + y k z k + x k z k = c .


Introduction.
Let p be an odd prime, let k be a fixed positive integer with 2k|p − 1, and let F p be the finite field with p elements.It is clear that there exists at least one generator of F p , and the number of all generators of F p is equal to φ(p − 1), where φ(n) is Euler's function.The main purpose of this paper is to study the following two problems: (A) for any element 0 ≠ c ∈ F p whether there exist three generators x, y, and z ∈ F p such that , k, p) denotes the number of all solutions of (1.1).
What can be said about the asymptotic properties of N(c, k, p)?In this paper, we use the estimates for general Gauss sums and the properties of Dirichlet characters to study the above two problems and prove the following main conclusion.
Theorem 1.1.Let p be an odd prime and k a fixed positive integer with 2k|p − 1.Then for any element 0 ≠ c ∈ F p , the asymptotic formula 1) , (1.2) where |θ| ≤ 54k 4 and ω(n) denotes the number of all distinct prime divisors of n.
From this theorem, we may immediately deduce the following corollary.
Corollary 1.2.Let p be a prime large enough and k a fixed positive integer with 2k|p −1.Then for any integer 1 ≤ c ≤ p −1, there exist three primitive roots x, y, and z modulo p such that the congruence (1.3) 2. Some lemmas.In this section, we give several lemmas which are necessary in the proof of Theorem 1.1.First, we let where χ denotes a Dirichlet character mod q, e(y) = e 2πiy .Then we have the following lemma.
Lemma 2.1.Let p be an odd prime and k a positive integer with k|p − 1.

Then for any integer
where χ 0 denotes the principal character mod p.
Proof.Let g be a fixed primitive root mod p, then for any integer n with p n, there exist two integers l and i such that n ≡ g lk+i (mod p), here 0 ≤ i < k.If b runs through a complete residue system mod p, then g l b also runs through a complete residue system mod p, so that we have (2.3)Note the trigonometric identity (2.4) Further, g i b k , i = 0,...,k − 1; b = 1,...,p runs through k complete residue systems mod p so that we have the identity otherwise.
Lemma 2.2.Let p be an odd prime and let n be an integer.Then , if n is a primitive root of p, 0, otherwise, (2.7) where ind n denotes the index of n relative to some fixed primitive root of p, µ(n) is the Möbius function, and k a=1 denotes the summation over all a such that (a, k) = 1.
Lemma 2.3.Let p be an odd prime, let k a fixed positive integer with 2k|p − 1, and let χ 1 , χ 2 , and χ 3 be three Dirichlet characters mod p. Then for any Proof.Let g be any fixed primitive root mod p, then, from the properties of primitive roots and reduced residue system mod p, we have (2.9) Let h be a fixed quadratic nonresidue modulo p, then (2.10) From (2.9) and (2.10), we have (2.11) Applying Lemma 2.1 to (2.11), we immediately get the estimate This proves Lemma 2.3.

Proof of the Theorem 1.1.
We only prove that Theorem 1.1 is true if F p is a complete residue system modulo p, then, from the isomorphism properties of the finite field, we can deduce that Theorem 1.1 is true for any finite field F p .Let p be an odd prime and Ꮽ(p) = Ꮽ denotes the set of all primitive roots modulo p in the interval [1,p − 1], then, from the trigonometric identity (2.4) and Lemma 2.2, we have First, we estimate the main term R 1 .Note (2.4) and p−1 a=1 χ(a) = 0 (χ is a nonprincipal character modulo p), from the definition of Dirichlet characters, we have From these identities and (3.2), we immediately get the main term In order to estimate the error term R 2 in (3.1), first we separate R 2 into four parts.That is, (3.5) Let g be any fixed primitive root mod p. Then note that 2k|p − 1, from the properties of primitive roots and reduced residue system mod p, we have (3.6) Applying Lemmas 2.1 and 2.3 to (3.6), we immediately get Using the same method of proving (3.7) and Lemma 2.3, and noting the identity 1) .
Note 3.1.Using the similar method of proving Theorem 1.1, we can also get the asymptotic formula 1) , (3.12) where

Journal of Applied Mathematics and Decision Sciences
Special Issue on Intelligent Computational Methods for Financial Engineering

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning u; a, j) = e(a ind u/j), χ(v; b, h) = e(b ind v/h), and χ(w; d, l) = e(d ind w/l) are three Dirichlet characters mod p.Since j > 1, h > 1, l > 1, and (b, h) = (a, j) = (d, l) = 1, the characters χ(u; a, j), χ(v; b, h), and χ(w; d, l)   are three primitive characters mod p.Therefore, we have

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation