On the System of Diophantine Equations x 2 − 6y 2 = −5 and x = az 2 − b

Mignotte and Pethö used the Siegel-Baker method to find all the integral solutions (x, y, z) of the system of Diophantine equations x 2 − 6y 2 = −5 and x = 2z 2 − 1. In this paper, we extend this result and put forward a generalized method which can completely solve the family of systems of Diophantine equations x 2 − 6y 2 = −5 and x = az 2 − b for each pair of integral parameters a, b. The proof utilizes algebraic number theory and p-adic analysis which successfully avoid discussing the class number and factoring the ideals.


Introduction
Let Z, N, and Q be the sets of all integers, positive integers, and rational numbers, and let , be the integers. The system of Diophantine equations 2 − 6 2 = −5, is a quartic model of an elliptic curve that has been investigated in many papers. Mignotte and Pethő [1] used the Siegel-Baker method to solve (1) for = 2 and = 1; however, their method was complicated as a combination of algebraic and transcendental number theory. In 1998, Cohn [2] gave an elementary proof of the above system of equations for = 2 and = 1. In 2004, Le [3] used a similar elementary method to extend the result of Cohn's work and proposed an effective method solving the system of equations where − 1 is the power of an odd prime. As an example, solutions of the equations for = 6 and = 8 are given in the paper so as to show the effectiveness of the method.
In this paper, we use algebraic number theory and Skolem's -adic method [4] to solve (1), and the method is relatively simple. In the proposed method, both the consideration of the class number in the field and the factorization of ideals of integral ring are avoided. Moreover, a faster algorithm proposed in [5] to compute the fundamental unit and the set of nonassociated factors is used.
In order to well interpret the main result, the symbol notation used in this paper is defined as below.
Here, we assume that ≥ 0 is an integer and 0 = 5+2 √ 6 is the fundamental unit in the field Q( √ 6), and let denote ( 0 + 0 )/( + ), where 2 = ± 0 or 2 = ± , with = 1 + √ 6; 2 denotes the conjugate of 2 in Q( √ 6). The main result of this paper is as follows. where is the fundamental unit of the totally complex quartic field of Q( ), is the nonassociated factor such that − = 4 ( − √−5), − denotes the relative conjugate of , and is referred to above.

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The Scientific World Journal As the application to the theorem, we give the following corollary.

Proof of the Theorem
Before the proof of the theorem, Lemma 3 is needed. Proof. Rewriting , we have Since 2 + 2 , 2 0 + 2 0 , and 0 + 0 are all rational integers and = √−5, then clearly is an algebraic number. Thus, the lemma is proven.

Proof of Theorem.
There are four separate cases in consideration during the proof. Since the process is very similar in each case, some details will be omitted for simplicity. Now we prove the theorem.
After rewriting the first equation of (1), factorization in the field Q( √ 6) yields Then we have Adding (5), we get The solution of (6) is split into four cases.
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Proof of the Corollary
Remark 4. The method of the proof of the corollary is a special instance of general procedure for the computation of integral points on some quartic model of elliptic curves. The method is relatively simple, because it avoids the use of class number and ideal factorization in imaginary quartic fields.
Before the proof, we give the following lemma.
Therefore, for > 1, the -adic valuation of the term of ( ) exceeds the -adic valuation of the term of ( 1 ). Thus, we complete the proof of Lemma 5.
To complete the proof of the corollary, the fundamental unit in the totally complex quartic field Q( ) is computed. Furthermore, nonassociated factor of 2 − 2√−5 in the ring of integers of Q( ) is also calculated. The idea of computation of fundamental unit and nonassociated factorization stems from Zhu and Chen [5,6] and Buchmann's work [7], which offered a fast implementation scheme. Results are obtained via MATHEMATICA 7.0, which are listed in Table 1.
For simplicity, we only list the positive -value of solutions ( , , ).