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

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 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.

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 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.

scholarly journals Positive solutions of the diophantine equation

1982 ◽  
Vol 5 (2) ◽  
pp. 311-314 ◽  
Author(s):  
W. R. Utz
Keyword(s):  
Positive Solutions ◽  
Diophantine Equation ◽  
Positive Integers ◽  
Integral Solutions

Integral solutions ofx3+λy+1−xyz=0are observed for all integralλ. Forλ=2the 13 solutions of the equation in positive integers are determined. Solutions of the equation in positive integers were previously determined for the caseλ=1.

scholarly journals On primitive solutions of the Diophantine equation x 2 + y 2 = M

2021 ◽  
Vol 19 (1) ◽  
pp. 863-868
Author(s):  
Chris Busenhart ◽  
Lorenz Halbeisen ◽  
Norbert Hungerbühler ◽  
Oliver Riesen
Keyword(s):  
Diophantine Equation ◽  
Gaussian Integers ◽  
Explicit Formulae ◽  
Integral Solutions

Abstract We provide explicit formulae for primitive, integral solutions to the Diophantine equation x 2 + y 2 = M {x}^{2}+{y}^{2}=M , where M M is a product of powers of Pythagorean primes, i.e., of primes of the form 4 n + 1 4n+1 . It turns out that this is a nice application of the theory of Gaussian integers.

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