Expressions for rational approximations to square roots of integers using Pell's equation

2019 ◽  
Vol 103 (556) ◽  
pp. 101-110
Author(s):  
Ken Surendran ◽  
Desarazu Krishna Babu
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Integer Solution ◽  
Rational Approximations ◽  
Positive Integer Solution ◽  
Pell’S Equation ◽  
Square Roots ◽  
Integer Solutions

There are recursive expressions (see [1]) for sequentially generating the integer solutions to Pell's equation:p2 −Dq2 = 1, whereDis any positive non-square integer. With known positive integer solutionp1 andq1 we can compute, using these recursive expressions,pnandqnfor alln> 1. See Table in [2] for a list of smallest integer, orfundamental, solutionsp1 andq1 forD≤ 128. These (pn,qn) pairs also formrational approximationstothat, as noted in [3, Chapter 3], match with convergents (Cn=pn/qn) of the Regular Continued Fractions (RCF, continued fractions with the numerator of all fractions equal to 1) for.

An upper bound for least solutions of the exponential Diophantine equation D1x2 - D2y2 = λkz

2015 ◽  
Vol 11 (04) ◽  
pp. 1107-1114 ◽  
Author(s):  
Hai Yang ◽  
Ruiqin Fu
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Pell’S Equation ◽  
Integer Solutions

Let D1, D2, D, k, λ be fixed integers such that D1 ≥ 1, D2 ≥ 1, gcd (D1, D2) = 1, D = D1D2 is not a square, ∣k∣ > 1, gcd (D, k) = 1 and λ = 1 or 4 according as 2 ∤ k or not. In this paper, we prove that every solution class S(l) of the equation D1x2-D2y2 = λkz, gcd (x, y) = 1, z > 0, has a unique positive integer solution [Formula: see text] satisfying [Formula: see text] and [Formula: see text], where z runs over all integer solutions (x,y,z) of S(l),(u1,v1) is the fundamental solution of Pell's equation u2 - Dv2 = 1. This result corrects and improves some previous results given by M. H. Le.

scholarly journals The Exponential Diophantine Equation2x+by=cz

2014 ◽  
Vol 2014 ◽  
pp. 1-3 ◽  
Author(s):  
Yahui Yu ◽  
Xiaoxue Li
Keyword(s):  
Positive Integer ◽  
Integer Solution ◽  
Elementary Approach ◽  
Positive Integer Solution ◽  
Integer Solutions ◽  
Positive Integers

Letbandcbe fixed coprime odd positive integers withmin{b,c}>1. In this paper, a classification of all positive integer solutions(x,y,z)of the equation2x+by=czis given. Further, by an elementary approach, we prove that ifc=b+2, then the equation has only the positive integer solution(x,y,z)=(1,1,1), except for(b,x,y,z)=(89,13,1,2)and(2r-1,r+2,2,2), whereris a positive integer withr≥2.

scholarly journals POSITIVE INTEGER SOLUTION OF THE MATRIX EQUATION Xn = A FOR THIRD-ORDER MATRICES IN THE CASE OF POSITIVE INTEGERS n

2018 ◽  
Vol 54 (2) ◽  
pp. 164-178
Author(s):  
K. L. Yakuto
Keyword(s):  
Positive Integer ◽  
Matrix Equation ◽  
Integer Solution ◽  
Third Order ◽  
Positive Integer Solution ◽  
The Third ◽  
Order Matrix ◽  
The Matrix ◽  
Integer Solutions ◽  
Different Order

The problem of the positive integer solution of the equation Xn = A for different-order matrices is important to solve a large range of problems related to the modeling of economic and social processes. The need to solve similar problems also arises in areas such as management theory, dynamic programming technique for solving some differential equations. In this connection, it is interesting to question the existence of positive and positive integer solutions of the nonlinear equations of the form Xn = A for different-order matrices in the case of the positive integer n. The purpose of this work is to explore the possibility of using analytical methods to obtain positive integer solutions of nonlinear matrix equations of the form Xn = A where A, X are the third-order matrices, n is the positive integer. Elements of the original matrix A are integer and positive numbers. The present study found that when the root of the nth degree of the third-order matrix will have zero diagonal elements and nonzero and positive off-diagonal elements, the root of the nth degree of the third-order matrix will have two zero diagonal elements and nonzero positive off-diagonal elements. It was shown that to solve the problem of finding positive integer solutions of the matrix equation for third-order matrices in the case of the positive integer n, the analytical techniques can be used. The article presents the formulas that allow one to find the roots of positive integer matrices for n = 3,…,5. However, the methodology described in the article can be adopted to find the natural roots of the third-order matrices for large n. 

scholarly journals A note on the Diophantine equation $|a^x-b^y|=c$

2010 ◽  
Vol 107 (2) ◽  
pp. 161
Author(s):  
Bo He ◽  
Alain Togbé ◽  
Shichun Yang
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Integer Solutions ◽  
Positive Integers

Let $a,b,$ and $c$ be positive integers. We show that if $(a,b) =(N^k-1,N)$, where $N,k\geq 2$, then there is at most one positive integer solution $(x,y)$ to the exponential Diophantine equation $|a^x-b^y|=c$, unless $(N,k)=(2,2)$. Combining this with results of Bennett [3] and the first author [6], we stated all cases for which the equation $|(N^k \pm 1)^x - N^y|=c$ has more than one positive integer solutions $(x,y)$.

Complete solutions of the simultaneous Pell’s equations x2 − (a2 − 1)y2 = 1 and y2 − pz2 = 1

2019 ◽  
Vol 15 (05) ◽  
pp. 1069-1074 ◽  
Author(s):  
Hai Yang ◽  
Ruiqin Fu
Keyword(s):  
Positive Integer ◽  
Diophantine Equations ◽  
Integer Solution ◽  
System Of Equations ◽  
Positive Integer Solution ◽  
Elementary Methods ◽  
Integer Solutions ◽  
Positive Integers ◽  
Complete Solutions

Let [Formula: see text] be a positive integer with [Formula: see text], and let [Formula: see text] be an odd prime. In this paper, by using certain properties of Pell’s equations and quartic diophantine equations with some elementary methods, we prove that the system of equations [Formula: see text] [Formula: see text] and [Formula: see text] has positive integer solutions [Formula: see text] if and only if [Formula: see text] and [Formula: see text] satisfy [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are positive integers. Further, if the above condition is satisfied, then [Formula: see text] has only the positive integer solution [Formula: see text]. By the above result, we can obtain the following corollaries immediately. (i) If [Formula: see text] or [Formula: see text], then [Formula: see text] has no positive integer solutions [Formula: see text]. (ii) For [Formula: see text], [Formula: see text] has only the positive integer solutions [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

scholarly journals A Note on Integer Solutions of the Diophantine Equation x2-dy2=1

1956 ◽  
Vol 3 (1) ◽  
pp. 55-56
Author(s):  
John Hunter
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Integer Solutions ◽  
Image Position

In the equationdis any positive integer which is not a perfect square. For convenience we shall consider only those solutions of (1) for which x and yare both positive. All the others can be obtained from these. In fact, it is well known that if (x0, y0) is the minimum positive integer solution of (1), then all integer solutions (x, y) are given byand, in particular, all positive integer solutions are given by

The exponential diophantine equation xy + yx = z2 via a generalization of the Ankeny–Artin–Chowla conjecture

2018 ◽  
Vol 14 (05) ◽  
pp. 1223-1228
Author(s):  
Hai Yang ◽  
Ruiqin Fu
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Class Number ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Pell Equation ◽  
Positive Integer Solution ◽  
Primitive Forms ◽  
Integer Solutions

Let [Formula: see text] be a positive integer which is not a square. Further, let [Formula: see text] be the least positive integer solution of the Pell equation [Formula: see text], and let [Formula: see text] denote the class number of binary quadratic primitive forms of discriminant [Formula: see text]. If [Formula: see text] satisfies [Formula: see text] and [Formula: see text], then [Formula: see text] is called an exceptional number. In this paper, under the assumption that there have no exceptional numbers, we prove that the equation [Formula: see text] has no positive integer solutions [Formula: see text] satisfy [Formula: see text] and [Formula: see text].

A NOTE ON JEŚMANOWICZ’ CONJECTURE CONCERNING NONPRIMITIVE PYTHAGOREAN TRIPLES

2020 ◽  
pp. 1-11
Author(s):  
YASUTSUGU FUJITA ◽  
MAOHUA LE
Keyword(s):  
Positive Integer ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Baker’S Method ◽  
Pythagorean Triples ◽  
Integer Solutions ◽  
Positive Integers

Abstract Jeśmanowicz conjectured that $(x,y,z)=(2,2,2)$ is the only positive integer solution of the equation $(*)\; ((\kern1.5pt f^2-g^2)n)^x+(2fgn)^y=((\kern1.5pt f^2+g^2)n)^x$ , where n is a positive integer and f, g are positive integers such that $f>g$ , $\gcd (\kern1.5pt f,g)=1$ and $f \not \equiv g\pmod 2$ . Using Baker’s method, we prove that: (i) if $n>1$ , $f \ge 98$ and $g=1$ , then $(*)$ has no positive integer solutions $(x,y,z)$ with $x>z>y$ ; and (ii) if $n>1$ , $f=2^rs^2$ and $g=1$ , where r, s are positive integers satisfying $(**)\; 2 \nmid s$ and $s<2^{r/2}$ , then $(*)$ has no positive integer solutions $(x,y,z)$ with $y>z>x$ . Thus, Jeśmanowicz’ conjecture is true if $f=2^rs^2$ and $g=1$ , where r, s are positive integers satisfying $(**)$ .

scholarly journals On the Diophantine Equation x2 – kxy + ky2 + ly = 0, l = 2n

2017 ◽  
Vol 55 (1) ◽  
pp. 115-118
Author(s):  
Sukrawan Mavecha
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Positive Integer Solution

AbstractWe consider the Diophantine equation x2-kxy+ky2+ ly = 0 for l = 2nand determine for which values of the odd integer k, it has a positive integer solution x and y.

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