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 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.

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 A GENERALIZATION OF THE RAMANUJAN–NAGELL EQUATION

2018 ◽  
Vol 61 (03) ◽  
pp. 535-544
Author(s):  
TOMOHIRO YAMADA
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solutions

AbstractWe shall show that, for any positive integer D > 0 and any primes p1, p2, the diophantine equation x2 + D = 2sp1kp2l has at most 63 integer solutions (x, k, l, s) with x, k, l ≥ 0 and s ∈ {0, 2}.

On D. J. Lewis's Equation x3+117y3 = 5

1971 ◽  
Vol 14 (1) ◽  
pp. 111-111 ◽  
Author(s):  
R. Finkelstein ◽  
H. London
Keyword(s):  
Diophantine Equation ◽  
Exact Number ◽  
Integer Solutions

In a recent publication [2], D. J. Lewis stated that the Diophantine equation x3+117y3 = 5 has at most 18 integer solutions, but the exact number is unknown. In this paper we shall solve this problem by proving the followingTheorem. The equationx3+117y3 = 5 has no integer solutions.

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