scholarly journals A GENERALISATION OF THE DIOPHANTINE EQUATION x^2+8∙7^b=y^2r

2021 ◽  
Vol 40 (2) ◽  
pp. 25-39
Author(s):  
Siti Hasana Sapar ◽  
Kai Siong Yow
Keyword(s):  
Diophantine Equation ◽  
Special Cases ◽  
Integral Solutions

We investigate the integral solutions to the Diophantine equation where . We first generalise the forms of and that satisfy the equation. We then show the general forms of non-negative integral solutions to the equation under several conditions. We also investigate some special cases in which the integral solutions exist.

The Diophantine equation x2+3 = yn

1993 ◽  
Vol 35 (2) ◽  
pp. 203-206 ◽  
Author(s):  
J. H. E. Cohn
Keyword(s):  
Diophantine Equation ◽  
Special Cases ◽  
Recent Origin ◽  
Positive Integers

Many special cases of the equation x2+C= yn where x and y are positive integers and n≥3 have been considered over the years, but most results for general n are of fairly recent origin. The earliest reference seems to be an assertion by Fermat that he had shown that when C=2, n=3, the only solutions are given by x = 5, y = 3; a proof was published by Euler [1]. The first result for general n is due to Lebesgue [2] who proved that when C = 1 there are no solutions. Nagell [4] generalised Fermat's result and proved that for C = 2 the equation has no solution other than x = 5, y = 3, n = 3. He also showed [5] that for C = 4 the equation has no solution except x = 2, y = 2, n = 3 and x = 11, y = 5, n = 3, and claims in [6] to have dealt with the case C = 5. The case C = -1 was solved by Chao Ko, and an account appears in [3], pp. 302–304.

scholarly journals THE DIOPHANTINE EQUATION A4 + 2δB2 = Cn

2010 ◽  
Vol 06 (02) ◽  
pp. 311-338 ◽  
Author(s):  
MICHAEL A. BENNETT ◽  
JORDAN S. ELLENBERG ◽  
NATHAN C. NG
Keyword(s):  
Diophantine Equation ◽  
Related Equation ◽  
Integral Solutions

In a previous paper, the second author proved that the equation [Formula: see text] had no integral solutions for prime p > 211 and (A,B,C) = 1. In the present paper, we explain how to extend this result to smaller exponents, and to the related equation [Formula: see text]

scholarly journals A Peer Search on Integer Solutions to Quadratic Diophantine Equation with Three Unknowns (Equation)

2020 ◽  
pp. 166-171
Author(s):  
A. Vijayasankar ◽  
Sharadha Kumar ◽  
M. A. Gopalan
Keyword(s):  
Diophantine Equation ◽  
Quadratic Diophantine Equation ◽  
Integer Solutions ◽  
Integral Solutions

The non- homogeneous ternary quadratic diophantine (Equation) is analyzed for its patterns of non-zero distinct integral solutions. Various interesting relations between the solutions and special numbers namely polygonal, Pronic and Gnomonic numbers are exhibited.

scholarly journals GENERATORS OF ARITHMETIC QUATERNION GROUPS AND A DIOPHANTINE PROBLEM

2010 ◽  
Vol 06 (06) ◽  
pp. 1311-1328
Author(s):  
MAJID JAHANGIRI
Keyword(s):  
Diophantine Equation ◽  
Quaternion Algebra ◽  
Discrete Subgroup ◽  
Pell’S Equation ◽  
Diophantine Problem ◽  
Empirical Results ◽  
Group Of Units ◽  
Integral Solutions ◽  
Mod P

Let p be a prime and a a quadratic non-residue ( mod p). Then the set of integral solutions of the Diophantine equation [Formula: see text] form a cocompact discrete subgroup Γp, a ⊂ SL(2, ℝ) which is commensurable with the group of units of an order in a quaternion algebra over ℚ. The problem addressed in this paper is an estimate for the traces of a set of generators for Γp, a. Empirical results summarized in several tables show that the trace has significant and irregular fluctuations which is reminiscent of the behavior of the size of a generator for the solutions of Pell's equation. The geometry and arithmetic of the group of units of an order in a quaternion algebra play a key role in the development of the code for the purpose of this paper.

On the integral solutions of the Diophantine equation x4 + y4 = 2kz3 where k > 1

2021 ◽  
Author(s):  
Shahrina Ismail ◽  
Kamel Ariffin Mohd Atan ◽  
Kai Siong Yow ◽  
Diego Sejas Viscarra
Keyword(s):  
Diophantine Equation ◽  
Integral Solutions

Laminar Natural Convection under Nonuniform Gravity

1972 ◽  
Vol 94 (1) ◽  
pp. 80-86 ◽  
Author(s):  
J. Lienhard ◽  
R. Eichhorn ◽  
V. Dhir
Keyword(s):  
Natural Convection ◽  
Solution Method ◽  
Leading Edge ◽  
General Integral ◽  
Special Cases ◽  
Method Of Solution ◽  
Arbitrary Position ◽  
Laminar Natural Convection ◽  
Position Dependence ◽  
Integral Solutions

Laminar natural convection is analyzed for cases in which gravity varies with the distance from the leading edge of an isothermal plate. The study includes situations in which gravity varies by virtue of the varying slope of a surface. A general integral solution method which includes certain known integral solutions as special cases is developed to account for arbitrary position-dependence of gravity. A series method of solution is also developed for the full equations. Although it is more cumbersome it provides verification of the integral method.

scholarly journals A note on the ternary Diophantine equation x 2 − y 2 m = z n

2021 ◽  
Vol 29 (2) ◽  
pp. 93-105
Author(s):  
Attila Bérczes ◽  
Maohua Le ◽  
István Pink ◽  
Gökhan Soydan
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Special Cases ◽  
Positive Integers

Abstract Let ℕ be the set of all positive integers. In this paper, using some known results on various types of Diophantine equations, we solve a couple of special cases of the ternary equation x2 − y2m = zn , x, y, z, m, n ∈ ℕ, gcd(x, y) = 1, m ≥ 2, n ≥ 3.

On the Diophantine Equation n(n + d) · · · (n + (k − 1)d) = byl

2004 ◽  
Vol 47 (3) ◽  
pp. 373-388 ◽  
Author(s):  
K. Győry ◽  
L. Hajdu ◽  
N. Saradha
Keyword(s):  
Arithmetic Progression ◽  
Diophantine Equation ◽  
Arithmetic Progressions ◽  
Rational Solutions ◽  
Initial Term ◽  
The Common ◽  
Integral Solutions

AbstractWe show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obláth for the case of squares, and an extension of a theorem of Győry on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in n, y when b = d = 1. We show that there are only finitely many solutions in n, d, b, y when k ≥ 3, l ≥ 2 are fixed and k + l > 6.

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