Counting Natural and Integer Solutions to Equations and Inequalities

2021 ◽  
Vol 67 (7) ◽  
pp. 21-25
Author(s):  
Aleksa Srdanov ◽  
Dragan Stojiljkovic
Keyword(s):  
Solutions To Equations ◽  
Integer Solutions

scholarly journals Monochromatic solutions to equations with unit fractions

1991 ◽  
Vol 43 (3) ◽  
pp. 387-392 ◽  
Author(s):  
Tom C. Brown ◽  
Voijtech Rödl
Keyword(s):  
Solutions To Equations ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

Our main result is that if G(x1, …, xn) = 0 is a system of homogeneous equations such that for every partition of the positive integers into finitely many classes there are distinct y1,…, yn in one class such that G(y1, …, yn) = 0, then, for every partition of the positive integers into finitely many classes there are distinct Z1, …, Zn in one class such thatIn particular, we show that if the positive integers are split into r classes, then for every n ≥ 2 there are distinct positive integers x1, x1, …, xn in one class such thatWe also show that if [1, n6 − (n2 − n)2] is partitioned into two classes, then some class contains x0, x1, …, xn such that(Here, x0, x2, …, xn are not necessarily distinct.)

scholarly journals On positive solutions to equations involving the one-dimensional p-Laplacian

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Ruyun Ma ◽  
Yanqiong Lu ◽  
Ahmed Omer Mohammed Abubaker
Keyword(s):  
Positive Solutions ◽  
Solutions To Equations ◽  
One Dimensional ◽  
The One

scholarly journals A closer look at some new identities

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).

scholarly journals Solution of certain Pell equations

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].

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

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 $60^\circ$.

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