On Consecutive Integer Solutions for y 2 - k = x 3

1975 ◽  
Vol 48 (2) ◽  
pp. 281 ◽  
Author(s):  
S. P. Mohanty
Keyword(s):  
Consecutive Integer ◽  
Integer Solutions

scholarly journals Diophantine equations of Erdös-Moser type

1996 ◽  
Vol 53 (2) ◽  
pp. 281-292 ◽  
Author(s):  
Pieter Moree
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Consecutive Integer ◽  
Elementary Method ◽  
Integer Solutions

Using an old result of Von Staudt on sums of consecutive integer powers, we shall show by an elementary method that the Diophantine equation 1k + 2k + … + (x − l)k = axk has no solutions (a, x, k) with k > 1, . For a = 1 this equation reduces to the Erdös-Moser equation and the result to a result of Moser. Our method can also be used to deal with variants of the equation of the title, and two examples will be given. For one of them there are no integer solutions with

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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