On Polynomials Solutions of Quadratic Diophantine Equations

A Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface or more general object, and ask about the lattice points on it. The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems Diophantus initiated is now called Diophantine analysis. A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations was an achievement of the twentieth century. For example, the equation is known the linear Diophantine equation. In general, the Diophantine equation is the equation given by

In this paper, we consider the number of polynomial solutions of Diophantine equation . We also obtain some formulas and recurrence relations on the polynomial solution   , n n X Y .E = 1 ax by 

Introduction
A Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only.Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations.In more technical language, they define an algebraic curve, algebraic surface or more general object, and ask about the lattice points on it.The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra.The mathematical study of Diophantine problems Diophantus initiated is now called Diophantine analysis.A linear Diophantine equation is an equation between two sums of monomials of degree zero or one.While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations was an achievement of the twentieth century.For example, the equation is known the linear Diophantine equation.In general, the Diophantine equation is the equation given by = 0 f  ax c The equation provided that is odd.There is no solution of other than  given by   where 1 1 x y is the least positive solution called the fundamental solution, which there are different method for finding it.The reader can find many references in the subject in the book [1].
We recall that there are many papers in which are considered different types of Pell's equation.and the equation and he obtained some formulas for its integer solutions.He mentioned two conjecture which was proved by A. S. Shabani [8].In [9], we extend the work in [2,7] by considering the Pell equation when be a positive non-square and , we obtain some formulas for its integer solutions.

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In [11,12], the number of integer solutions of Diophantine equation and Diophantine equation over is considered, where

Let be a polynomial in
In this paper, we consider the number of polynomial solutions of Diophantine equation 4 4 We also obtain some formulas and recurrence relations on the polynomial solution Note that the resolution of in its present form is difficult, that is, we can not determine how many solutions has and what they are.So, we have to transform into an appropriate Diophantine equation which can be easily solved.To get this let be a translation for some H and By applying the transformation to we get and we have the Diophantine equation which is a Pell equation.

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Now, we try to find all polynomial solutions T E and then we can retransfer all results from to by using the inverse of Theorem 2.1: Let be the Diophantine equation in (3), then 1) The fundamental solution of is 3) The solutions satisfy the recurrence relations U V can be given by 5) The -th solution 1) It is easily seen that is the fundamental polynomial solution of since 2) We prove it using the method of mathematical induction.Let , by ( 5) we get which is the fundamental solution and so is a solution of .Now, we assume that the Diophantine equation ( 4) is satisfied for that is We try to show that this equation is also satisfied for Applying (5), we find that Hence, we conclude that So is also solution of 3) Using (9), we find that for 4) We prove it using the method of mathematical induction.For we get Let us assume that this relation is satisfied for that is, Then using ( 9) and ( 10), we conclude that Similarly, we prove that We prove it using the method of mathematical induction.For , we have and we show that it holds for U sing (6) , we have could be transformed into the Diophantine equation via the transformation T Also, we showed that So, we can retransfer all results from to by applying the inverse of Thus, we can give the following main theorem: Theorem 2.2: Let be the Diophantine equation in (1).Then 1) The fundamental (minimal) solution of is completing the proof.As we reported above, the Diophantine equation X Y satisfy the recurrence relations (see ( 11)) 4) The solutions   , n n X Y satisfy the recurrence relations (see ( 12))

Acknowledgements
We would like to thank Saäd Chandoul and Massöuda Loörayed for helpful discussions and many remarks.