scholarly journals Integral Solutions of the Ternary Cubic Equation 6(x^2+y^2 )-11xy=288z^3

2022 ◽  
Vol 10 (1) ◽  
pp. 62-65
Author(s):  
C. Saranya
Keyword(s):  
Infinite Number ◽  
Diophantine Equation ◽  
Cubic Equation ◽  
Zero Integral ◽  
Integral Solutions

Abstract: The Ternary cubic Diophantine Equation represented by૟(࢞ ࢟ + ૛ ࢠૡૡ = ૛࢟࢞૚૚ − (૛ ૜ is analyzed for its infinite number of non-zero integral solutions. A few interesting among the solutions are also discussed. Keywords: Diophantine equation, Integral solutions, cubic equation with three unknowns, Ternary equation.

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 Observations on the Ternary Quadratic Equation X2=24α2+Y2

2014 ◽  
Vol 10 ◽  
pp. 59-61
Author(s):  
M.A. Gopalan ◽  
V. Sangeetha ◽  
Manju Somanath
Keyword(s):  
Quadratic Equation ◽  
Zero Integral ◽  
Polygonal Numbers ◽  
Integral Solutions

The Ternary Quadratic Equation X2=24α2+Y2 is Considered.Employing its Non-Zero Integral Solutions, Relations among a few Special Polygonal Numbers are Determined.

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 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 LATTICE POINTS ON THE HOMOGENEOUS CUBIC EQUATION WITH FOUR UNKNOWNS x2 - xy + y2 + 4w2 = 8z3

2020 ◽  
Vol 8 (8) ◽  
Author(s):  
Manju Somanath ◽  
Radhika Das ◽  
V.A Bindu
Keyword(s):  
Three Dimensional ◽  
Lattice Points ◽  
Cubic Equation ◽  
Figurate Number ◽  
Integral Solutions

The Homogeneous cubic equation with four unknowns represented by the equation x2 - xy + y2 + 4w2 = 8z3  is analyzed for its patterns of non zero distinct integral solutions. Here we exhibit four different patterns. In each pattern we can find some interesting relations between the solutions and special numbers like Polygonal number, Three-Dimensional Figurate number, Star number, Rhombic Dodecahedral number etc.

Sign in / Sign up

Export Citation Format

Share Document