Integral Solutions of the Binary Quadratic Diophantine Equation x2 - 2xy -y2 + 2x +14y= 72

2021 ◽  
pp. 83-90
Author(s):  
C. Saranya ◽  
G. Janaki
Keyword(s):  
Diophantine Equation ◽  
Quadratic Diophantine Equation ◽  
Integral Solutions

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.

Coordinates for vertices of regular honeycombs in hyperbolic space

1966 ◽  
Vol 293 (1432) ◽  
pp. 94-107 ◽  
Keyword(s):  
Symmetry Group ◽  
Hyperbolic Space ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Hyperbolic Spaces ◽  
Fundamental Region ◽  
Homogeneous Coordinates ◽  
Quadratic Diophantine Equation ◽  
Integral Solutions

Schlegel (1883) enumerated all regular honeycombs of hyperbolic spaces of three or more dimensions, having finite cells and vertex figures. Coxeter (1954) extended this enumeration to include honeycombs with infinite cells and/or infinite vertex figures, the fundamental region of the symmetry group still being finite. One of these, {4, 4, 3}, was shown to have for its vertices the points whose coordinates are proportional to the integral solutions of a quadratic Diophantine equation (Coxeter & Whitrow 1950). In the present paper, certain quadratic Diophantine equations are found whose solutions provide homogeneous coordinates for the vertices of {6, 3, 3}, {6, 3, 4}, {4, 3, 4, 3} and {3, 4, 3, 3, 3}. A method is also given for finding coordinates for the vertices of the remaining honeycombs (with finite vertex figures), and the simplest of these are listed.

scholarly journals On a class of insoluble binary quadratic diophantine equations

1991 ◽  
Vol 123 ◽  
pp. 141-151 ◽  
Author(s):  
Franz Halter-Koch
Keyword(s):  
Diophantine Equation ◽  
Class Number ◽  
Diophantine Equations ◽  
Number Fields ◽  
Quadratic Number ◽  
Number Problem ◽  
Quadratic Number Fields ◽  
Quadratic Diophantine Equation ◽  
Real Quadratic

The binary quadratic diophantine equationis of interest in the class number problem for real quadratic number fields and was studied in recent years by several authors (see [4], [5], [2] and the literature cited there).

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 ON THE TERNARY QUADRATIC DIOPHANTINE EQUATION 6z2 = 6x2 – 5y2

2020 ◽  
Vol 1 (1) ◽  
pp. 14-22
Author(s):  
M. Gopalan ◽  
S. Nandhini ◽  
J. Shanthi
Keyword(s):  
Diophantine Equation ◽  
Quadratic Equation ◽  
Quadratic Diophantine Equation ◽  
Integer Solutions ◽  
Distinct Integer ◽  
The Given

The ternary homogeneous quadratic equation given by 6z2 = 6x2 -5y2 representing a cone is analyzed for its non-zero distinct integer solutions. A few interesting relations between the solutions and special polygonal and pyramided numbers are presented. Also, given a solution, formulas for generating a sequence of solutions based on the given solutions are presented.

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