The Polynomial Solutions of Quadratic Diophantine Equation  = 0

Abstract

In this study, we consider the number of polynomial solutions of the Pell equation is formulated for a nonsquare polynomial using the polynomial solutions of the Pell equation . Moreover, a recurrence relation on the polynomial solutions of the Pell equation . Then, we consider the number of polynomial solutions of Diophantine equation . We also obtain some formulas and recurrence relations on the polynomial solution of .

1. Introduction

A Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations with unknown variables and involve finding integers that work correctly for all equations. The equation is known as the liner Diophantine equation. In general, the Diophantine equation is the equation given by

The equation , with given integers and and unknowns and , is called Pell’s equation. The most interesting case of the equation arises when be a positive nonsquare. Pell’s equation was solved by Lagrange in terms of simple continued fractions. We recall that there are many studies in which there are different types of Pell’s equation. Many authors such as Tekcan [1], Matthews [2], Chandoul [3], and Li [4] have researched. In [5], the equation was considered, and some formulas of its integer solutions were obtained. In [6, 7], the number of integer solutions of Diophantine equation and Diophantine equation over is considered, where . In [3, 8], the number of polynomial solutions of Diophantine equationand Diophantine equationover was considered, where be a polynomial in .

2. Preliminaries

In this section, we introduce the objects we need later and collect some important facts about them.

In [4], Li proved that the Pell equation has infinitely positive solutions. If is the fundamental solution, then for , . The pairs are all the positive solutions of the Pell equation. The ’s and ’s are strictly increasing to infinity and satisfy the recurrence relations:

Theorem 1. (Tekcan (see [5])). Let be the fundamental solution of the Pell equation and be the fundamental solution of the Pell equation . Then, the other solutions of the Pell equation are :(1)For ,(2)For ,(3)For ,

3. New Results

Our principal result is the following.

Theorem 2. Let be the fundamental solution of the Pell equation:Then, the other solutions of the Pell equation are , where(1)For ,(2)For ,(3)For ,

Proof. (1)We prove it using the method of mathematical induction. Let , and we get , which is the fundamental solution of equation (8). Now, we assume that (9) is satisfied for , that is, We try to show that this equation is also satisfied for . Applying (9), we find that Hence, we conclude that So, is also solution of equation (8).(2)Using (13), we find that For .(3)We prove it using the method of mathematical induction. For , we getHence,So, is satisfied for . Let us assume that this relation is satisfied for , that is,then, using (13) and (18), we conclude thatcompleting the proof.
Similarly, we prove thatNow, we give a relation between and .

Theorem 3. If be the fundamental solution of the equation then be a solution of the equation (8) and

Proof. Hence, it is easily seen thatsince is the fundamental solution of the equation , i.e, .

Theorem 4. Let be the fundamental solution of the equation (21) and be the fundamental solution of the equation (8). Then,(1)(2)(3)The solution satisfies the recurrence relations(4)The solution satisfies the recurrence relations

Proof. (1)From (24), we get Hence, it is easily seen that Since , and .(2)From (24), we get Then, Then, On the other hand, by using (10) and (32), we get Applying (32) and (33), we find Similarly, we prove that(3)From (25), we get Similarly, we prove that(4)From (22) and (26), we getSimilarly, we prove thatand we consider the number of polynomial solutions of Diophantine equation. Now,where is the fundamental solution of equation (21).
We have to transform into an appropriate Diophantine equation which can be easily solved. To get this, letbe a translation for some and .
By applying the transformation to , we getIn (42), we obtain and . So we get and . Consequently, for , we have the Diophantine equationwhich is Pell equation.
Now, we try to find all polynomial solutions of , and then, we can retransfer all results from to by using the inverse of .

Theorem 5. Let be the Diophantine equation in (40), where is the fundamental solution of equation (43); then,(1)The fundamental (minimal) solution of is(2)Define the sequencewhere is defined in (24). Then, is a solution of . So, it has infinitely many integer solutions .(3)The solutions satisfy the recurrence relations For .(4)The solutions satisfy the recurrence relations For .(5)The solutions satisfy the recurrence relations For .

Example 1. Let ; then, is the fundamental solution ofBy Theorem 3, is the fundamental solution ofBy Theorem 4, it is easily seen that some other solutions of equation areIt can be concluded now that the fundamental solution ofis . By Theorem 5, it is easily seen that some other solutions of equation are

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank Ahmad Issa for helpful discussion.