Pell Equation Research Articles

COMPLETE CLASSIFICATION OF QUADRATIC IRRATIONALS WITH PERIOD TWO

This article presents a comprehensive investigation into the classification of quadratic irrationals with period two in their continued fraction representations. Building upon foundational results in Number Theory, particularly in the context of continued fractions and Pell's equation, the study reveals intricate relationships between quadratic irrationals and their periodic structures. The main object of study is √N and properties of its continued fractions. While it is well-known that continued fractions of √N is periodic with periodic part being palindrome, the distribution of the lengths of the periodic parts are far from being complete. Our main goal will be to focus on the period two case and provide a complete characterization. The research's proved theorems clarify the conditions under which the period length is exactly two and give an insight into the underlying algebraic features. Additionally, it delves deeper by offering numerical analysis and illustrations demonstrating the distribution of period lengths among quadratic irrationals. This research opens up new paths for future studies on quadratic irrationals and how they're shown as continued fractions.

Extremal fixed points and Diophantine equations

The coupling constants of fixed points in the ϵ expansion at one loop are known to satisfy a quadratic bound due to Rychkov and Stergiou. We refer to fixed points that saturate this bound as extremal fixed points. The theories which contain such fixed points are those which undergo a saddle-node bifurcation, entailing the presence of a marginal operator. Among bifundamental theories, a few examples of infinite families of such theories are known. A necessary condition for extremality is that the sizes of the factors of the symmetry group of a given theory satisfy a specific Diophantine equation, given in terms of what we call the extremality polynomial. In this work we study such Diophantine equations and employ a combination of rigorous and probabilistic estimates to argue that these infinite families constitute rare exceptions. The Pell equation, Falting’s theorem, Siegel’s theorem, and elliptic curves figure prominently in our analysis. In the cases we study here, more generic classes of multi-fundamental theories saturate the Rychkov-Stergiou bound only in sporadic cases or in limits where they degenerate into simpler known examples.

Some new results on the negative polynomial Pell’s equation

We consider the negative polynomial Pell's equation $P^2(X)-D(X)Q^2(X)=-1$, where $D(X)\in \mathbb{Z}[X]$ be some fixed, monic, square-free, even degree polynomials. In this paper, we investigate the existence of polynomial solutions $P(X), Q(X)$ with integer coefficients.

Correction to: On the Euler Function of Y-Coordinates of Pell Equations and Repdigits

Correction to: On the Euler Function of Y-Coordinates of Pell Equations and Repdigits

The Integer Solution of the Negative Pell Equation Involving Palindrome Numbers

Abstract: The objective of the article is to find an integer solution that is non-zero for the negative Pell’s equation involving Palindrome number

Bounds for the smallest integral solutionof Pell equation over a number field

Bounds for the smallest integral solutionof Pell equation over a number field

On the x-coordinates of Pell equations that are products of two Pell numbers

Abstract Let (Pm ) m≥0 be the sequence of Pell numbers given by P 0 = 0, P 1 = 1, and P m+2 = 2P m+1 + Pm for all m ≥ 0. In this paper, for an integer d ≥ 2 which is square free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 − dy 2 = ± 1, which is a product of two Pell numbers.

Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers

The classical problem of finding all integers a and M such that the sums of M consecutive squared integers a+i2 equal the squared integer s2, where M is the number of terms in the sum, a2 the first term and a≥1, 0≤i≤M−1, yields remarkable regular linear features when plotting the values of M as a function of a. These linear features correspond to groupings of pairs of a values for successive same values of M found on either side of straight lines of equation μM=2a+c, where c is an integer constant and μ a parameter taking some rational values, called allowed values. We find expressions of a and s as a function of M for the allowed values of μ and M and parametric expressions of a, M, and s. Further, Pell equations deduced from the conditions of M are solved to find the allowed values of μ and to provide all solutions in a and M. These results yield new insights into the overall properties of the classical problem of the sums of consecutive squared integers equal to squared integers and allow us to solve this problem completely by providing all solutions in infinite families.

Quartic integral polynomial Pell equations

In this paper we classify all monic, quartic, polynomials d(x)∈Z[x] for which the Pell equationf(x)2−d(x)g(x)2=1 has a non-trivial solution with f(x),g(x)∈Z[x].

A counterexample to the Pellian equation conjecture of Mordell

A counterexample to the Pellian equation conjecture of Mordell

On the X-coordinates of Pell equations X2 − dY2 = ±1 as difference of two Fibonacci numbers

In this paper, we show that there is at most one value of the positive integer [Formula: see text] participating in the Pell equation [Formula: see text], which is a difference of two Fibonacci numbers.

On Diophantine equation xb=Dy

Diophantine equation is an important part of the number theory and it has been widely studied for a long time. There are many studies on solving the integral solutions of Diophantine equations using algebraic methods. This paper uses documentation method, sums up the results of research in different documents of finding the integral solutions of some Diophantine equations in various conditions, especially the utilization of theories on Pell equation, which in the form of xb=Dy. This paper mainly considers the situations when a =2 or a=3, which is a particular type of Diophantine equation. Several of the studies focus on the same equation but using different ways. Also, some subsequent studies on different equations are based on the former theorems provided by other writers and expand these theorems to broader applications. In order to emphasize the theorems obtained by the essays quoted and the methods the authors used, most of the mathematical procedures in their proofs are omitted.

Solutions of Some Quadratic Diophantine Equations

Let $P(t)_i^{\pm}=t^{2k} \pm i t^m$ be a non square polynomial and $Q(t)_i^{\pm}=4k^2t^{4k-2}+i^2m^2t^{2m-2} \pm 4imkt^{2k+m-2} -4t^{2k} \mp 4it^m -1$ be a polynomial, such that $k \geq 2m$ and $i \in \left\lbrace 1,2 \right\rbrace $. In this paper, we consider the number of integer solutions of Diophantine equation $$E\ :\ x^2-P(t)_i^{\pm}y^2-2P'(t)_i^{\pm}x+4 P(t)_i^{\pm} y +Q(t)_i^{\pm}=0.$$ We extend a previous results given by A. Tekcan and A. Chandoul et al. . We also derive some recurrence relations on the integer solutions of a Pell equation.

Negative Pell Equation and Stationary Configurations of Point Vortices on the Plane

Negative Pell Equation and Stationary Configurations of Point Vortices on the Plane

On Y-coordinates of Pell equations which are Fibonacci numbers

Let d ge 2 be an integer which is not a square. We show that if (F_n)_{nge 0} is the Fibonacci sequence and (X_m, Y_m)_{mge 1} is the mth solution of the Pell equation X^2 -dY^2 = pm 1, then the equation Y_m = F_n has at most two positive integer solutions (m, n) except for d=2 when it has three solutions (m,n)=(1,2),(2,3),(3,5).

Ambiguous solutions of a Pell equation

Ambiguous solutions of a Pell equation

An Integral Solution of Negative Pell Equation

Abstract: We look for non–trivial integer solution to the equation

On the cubic Pell equation over finite fields

The classical Pell equation can be extended to the cubic case considering the elements of norm one in , which satisfy The solution of the cubic Pell equation is harder than the classical case, indeed a method for solving it as Diophantine equation is still missing [3]. In this paper, we study the cubic Pell equation over finite fields, extending the results that hold for the classical one. In particular, we provide a novel method for counting the number of solutions in all possible cases depending on the value of r. Moreover, we are also able to provide a method for generating all the solutions.

Proofs of Chappelon and Ramírez Alfonsín Conjectures On Square Frobenius Numbers and their Relationship to Simultaneous Pell Equations

Proofs of Chappelon and Ramírez Alfonsín Conjectures On Square Frobenius Numbers and their Relationship to Simultaneous Pell Equations

SOLUTIONS OF PELL'S EQUATION INVOLVING SOPHIE GERMAIN PRIMES

We bring forth one of the most sought after and intriguing space pertaining to the magical world of Number Theory; and our attempts to uncover the continuing research and developments to find solutions for different aspects of the Pells equation. As indicated in this research paper, we attempt to find the possible solutions for the Pells equation x2 = 41y2-5m for all choice of m ∈ N. In this paper, we focused primarily on Pell's equations involving the Sophie Germain primes and present to you another mysterious series and pattern typically associated with the Pells equation. As we proceed through the research, we will bring to the fore the recurrence relations among the identied solutions.