On the Odd Prime Solutions of the Diophantine Equationxy+yx=zz

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

International Journal of Number Theory ◽

10.1142/s1793042115500608 ◽

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 ◽

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.

On the solvability of the simultaneous Pell equations x2 − ay2 = 1 and y2 − bz2 = v12

International Journal of Number Theory ◽

10.1142/s1793042121500731 ◽

2021 ◽

pp. 1-12

Author(s):

Ruiqin Fu ◽

Hai Yang

Keyword(s):

Positive Integer ◽

Diophantine Equations ◽

Prime Divisors ◽

Positive Integer Solution ◽

Pell Equations ◽

Integer Solutions ◽

Least Solution ◽

Simultaneous Pell Equations ◽

Necessary And Sufficient ◽

Positive Integers

Let [Formula: see text] be fixed positive integers such that [Formula: see text] is not a perfect square and [Formula: see text] is squarefree, and let [Formula: see text] denote the number of distinct prime divisors of [Formula: see text]. Let [Formula: see text] denote the least solution of Pell equation [Formula: see text]. Further, for any positive integer [Formula: see text], let [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations [Formula: see text] and [Formula: see text] to have positive integer solutions [Formula: see text] is obtained. By this result, we prove that if [Formula: see text] has a positive integer solution [Formula: see text] for [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, then [Formula: see text] and [Formula: see text], where [Formula: see text] is a positive integer, [Formula: see text] or [Formula: see text] and [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, [Formula: see text] is the integer part of [Formula: see text], except for [Formula: see text]

On the Diophantine System $x^2 - Dy^2 = 1-D$ and $x=2z^2-1$

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-14455 ◽

2004 ◽

Vol 95 (2) ◽

pp. 171 ◽

Author(s):

Maohua Le

Keyword(s):

Positive Integer ◽

Prime Power ◽

System Of Equations ◽

Elementary Method ◽

Diophantine System ◽

All Solutions ◽

Let $D$ be a positive integer such that $D-1$ is an odd prime power. In this paper we give an elementary method to find all positive integer solutions $(x, y, z)$ of the system of equations $x^2-Dy^2=1-D$ and $x=2z^2-1$. As a consequence, we determine all solutions of the equations for $D=6$ and $8$.

A closer look at some new identities

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171298000805 ◽

1998 ◽

Vol 21 (3) ◽

pp. 581-586

Author(s):

Geoffrey B. Campbell

Keyword(s):

Positive Integer ◽

Infinite Products ◽

Integer Solutions ◽

Origin Point

We obtain infinite products related to the concept of visible from the origin point vectors. Among these is∏k=3∞(1−Z)φ,(k)/k=11−Zexp(Z32(1−Z)2−12Z−12Z(1−Z)), |Z|<1,in whichφ3(k)denotes for fixedk, the number of positive integer solutions of(a,b,k)=1wherea<b<k, assuming(a,b,k)is thegcd(a,b,k).

Solution of certain Pell equations

Asian-European Journal of Mathematics ◽

10.1142/s1793557118500560 ◽

2018 ◽

Vol 11 (04) ◽

pp. 1850056 ◽

Author(s):

Zahid Raza ◽

Hafsa Masood Malik

Keyword(s):

Positive Integer ◽

Fundamental Solution ◽

Positive Solutions ◽

Continued Fraction ◽

Pell Equation ◽

Lucas Sequences ◽

Pell Equations ◽

Integer Solutions ◽

Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

On the Diophantine equations z^2 = f(x)^2 ± f(x)f(y) + f(y)^2

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2021.27.2.88-100 ◽

2021 ◽

Vol 27 (2) ◽

pp. 88-100

Author(s):

Qiongzhi Tang ◽

Keyword(s):

Positive Integer ◽

Diophantine Equations ◽

Pell Equation ◽

Intersection Angle ◽

Integer Solutions ◽

Two Sides

Using the theory of Pell equation, we study the non-trivial positive integer solutions of the Diophantine equations $z^2=f(x)^2\pm f(x)f(y)+f(y)^2$ for certain polynomials f(x), which mean to construct integral triangles with two sides given by the values of polynomials f(x) and f(y) with the intersection angle $120^\circ$ or $60^\circ$.

A GENERALIZATION OF THE RAMANUJAN–NAGELL EQUATION

Glasgow Mathematical Journal ◽

10.1017/s0017089518000344 ◽

2018 ◽

Vol 61 (03) ◽

pp. 535-544

Author(s):

TOMOHIRO YAMADA

Keyword(s):

Positive Integer ◽

Diophantine Equation ◽

AbstractWe shall show that, for any positive integer D > 0 and any primes p1, p2, the diophantine equation x2 + D = 2sp1kp2l has at most 63 integer solutions (x, k, l, s) with x, k, l ≥ 0 and s ∈ {0, 2}.

The Positive Integer Solutions of a Diophantine Equation

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.713-715.1483 ◽

2015 ◽

Vol 713-715 ◽

pp. 1483-1486

Author(s):

Yi Wu ◽

Zheng Ping Zhang

Keyword(s):

Positive Integer ◽

Diophantine Equation ◽

Pell Equation ◽

Integer Solutions ◽

Basic Equations

In this paper, we studied the positive integer solutions of a typical Diophantine equation starting from two basic equations including a Diophantine equation and a Pell equation, and we will prove all the positive integer solutions of the typical Diophantine equation.

ON THE NUMBER OF SOLUTIONS OF THE DIOPHANTINE EQUATION axm−byn=c

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972709001002 ◽

2010 ◽

Vol 81 (2) ◽

pp. 177-185 ◽

Author(s):

BO HE ◽

ALAIN TOGBÉ

Keyword(s):

Positive Integer ◽

Diophantine Equation ◽

Number Of Solutions ◽

Integer Solutions ◽

Image Position ◽

Positive Integers

AbstractLet a, b, c, x and y be positive integers. In this paper we sharpen a result of Le by showing that the Diophantine equation has at most two positive integer solutions (m,n) satisfying min (m,n)>1.