Related papers: Solution of Certain Pell Equations

Let $\ k$ be a natural number and $d=k^{2}\pm 4$ or $k^{2}\pm 1$. In this paper, by using continued fraction expansion of $\sqrt{d},$ we find fundamental solution of the equations $x^{2}-dy^{2}=\pm 1$ and we get all positive integer…

In this study, we find continued fraction expansion of sqrt(d) when d=a^2+2a where a is positive integer. We consider the integer solutions of the Pell equation x^2-(a^2+2a)y^2=N when N={-1,+1,-4,+4}. We formulate the n-th solution…

In this study, we find continued fraction expansion of sqrt(d) when d=a^2b^2-b and d=a^2b^2-2b where a and b are positive integers. We consider the integer solutions of the Pell equations x^2-(a^2b^2-b)y^2=N and x^2-(a^2b^2-2b)y^2=N when N…

Let $ \{P_{n}\}_{n\geq 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 = P_2=1$ and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\geq 0 $. In this paper, we find all positive square-free integers $ d $ such that the Pell…

The Pell equation $x^2 - Dy^2 = 1$ with non-square $D > 1$ has infinitely many integer solutions, yet most research has centered on the asymptotic behavior of fundamental units as $D$ varies. By contrast, the exact distribution of solutions…

For an integer $k\geq 2$, let $\{F^{(k)}_{n}\}_{n\geqslant 2-k}$ be the $ k$--generalized Fibonacci sequence which starts with $0, \ldots, 0,1$ (a total of $k$ terms) and for which each term afterwards is the sum of the $k$ preceding terms.…

First, we consider the equation $ax^2 - by^2 + c = 0$, with $a,b \in N*$ and $c \in Z*$, which is a generalization of Pell's equation. Here, we show that: if this equation has an integer solution and $ab$ is not a perfect square, then it…

Two theorems are demonstrated giving analytical expressions of the fundamental solutions of the Pell equation $X^{2}-DY^{2}=1$ found by the method of continued fractions for two squarefree polynomial expressions of radicands of…

It is a generalization of Pell's equation $x^2-Dy^2=0$. Here, we show that: if our Diophantine equation has a particular integer solution and $ab$ is not a perfect square, then the equation has an infinite number of solutions; in this case…

For an integer $d\geq 2$ which is not a square, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-dY^2=\pm 1$ which is a Tribonacci number, with a few exceptions that we completely…

Let $ \{L_n\}_{n\ge 0} $ be the sequence of Lucas numbers given by $ L_0=2, ~ L_1=1 $ and $ L_{n+2}=L_{n+1}+L_n $ for all $ n\ge 0 $. In the first paper, for an integer $d\geq 2$ which is square-free, we show that there is at most one value…

We consider the simultaneous Pell equations $$x^2 - ay^2 = 1, \qquad z^2 - bx^2 = 1,$$ where $a > b\geq 2$ are positive integers. We describe a procedure which, for any fixed $b$, either confirms that the simultaneous Pell equations have at…

Let $b\ge 2$ be a given integer. In this paper, we show that there only finitely many positive integers $d$ which are not squares, such that the Pell equation $X^2-dY^2=1$ has two positive integer solutions $(X,Y)$ with the property that…

This short paper is concerned with polynomial Pell equations \[P^2-DQ^2=1,\] with $P,Q,D\in\Bbb C[X]$ and ${deg}(D)=2$. The main result shows that the polynomials $P$ and $Q$ are closely related to Chebyshev polynomials. We then investigate…

On the coordinate plane, the slopes $a$ and $b$ of two straight lines and the slope $c$ of one of their angle bisectors satisfy the equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$ Recently, an explicit formula for nontrivial integral solutions…

The polynomial Pell equation is \[P^2 - D Q^2 = 1\] where $D$ is a given integer polynomial and the solutions $P, Q$ must be integer polynomials. A classical paper of Nathanson \cite{Nat} solved it when $D(x) = x^2 + d$. We show that the…

For each positive integer $n$ it is shown how to construct a finite collection of multivariable polynomials $\{F_{i}:=F_{i}(t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor})\}$ such that each positive integer whose squareroot has a continued…

The purpose of this article is to give the solutions of the inverse problem for Pellian equations. For any rational number $0< a/b < 1$, the fundamental discriminants $D$ satisfying $(\lfloor \sqrt{D} \rfloor b + a)^2 - D b^2 = 4$ are given…

By applying methods recently developed by A. Smith with regards to Goldfeld's conjecture, we show that the density of square-free integers $d$ in $[1, N]$ for which the negative Pell equation $x^2 - dy^2 = -1$ has a solution is as predicted…

D. S. Hong and P. Pongsriiam have provided a necessary and sufficient condition for the generating function for Fibonacci numbers (resp. the Lucas numbers) to be an integer value, for rational numbers. In other words, their results relate…