International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395 -0056
Volume: 04 Issue: 03 | Mar -2017
p-ISSN: 2395-0072
www.irjet.net
OBSERVATIONS ON X2 + Y2 = 2Z2 - 62W2 G.Janaki* and R.Radha* *Department of Mathematics, Cauvery college for women, Trichy ---------------------------------------------------------------------***--------------------------------------------------------------------1), (-y, y, ±1, -y) satisfy the equation under
Abstract - The Quadratic equation with four unknowns of the form X2 + Y2 = 2Z2 - 62W2 has been studied for its
consideration. In the above quadruples, any two values
non-trivial distinct integral solutions. A few interesting
of the unknowns are the same. In [13], the Quadratic
relations among the solutions and special polygonal
equation with four unknowns of the form XY +X+ Y+1 =
numbers are presented.
Z2 -W2 has been studied for its non-trivial distinct integral solutions. Thus, towards this end we search for
Key Words: Quadratic equation with four unknowns,
non-zero distinct integral solutions of the equation
Integral solutions.
under consideration. A few interesting relations among
NOTATIONS:
the solutions are presented.
n( n 1) Triangular number of rank n. 2 n(5n 3) T7,n Heptagonal number of rank n. 2 T10,n n (4n 3) Decagonal number of rank n.
METHOD OF ANALYSIS:
T3,n
The Quadratic Diophantine equation with four unknowns under consideration is
T12,n n(5n 4) Dodecagonal number of rank n. T13,n T17,n T18,n
X2 + Y2 = 2Z2 - 62W2
n(11n 9) Tridecagonal number of rank n. 2 n(15n 13) Heptadecagonal number of rank n. 2 n(8n 7) Octadecagonal number of rank n.
(1)
The substitution of the linear transformations X= u + v, Y = v - u and W = u
(2)
in (1) leads to
Gnon (2n 1) Gnomonic number of rank n.
Z 2= 32 u2+v2
INTRODUCTION:
(3)
Four different choices of solutions to (3) are presented
In [ 1 to 12] some special types of quadratic
below. Once the values of u and v are known , using (2),
equations with four unknowns have been analyzed for
the corresponding values of X and Y are obtained.
their non-trivial integral solutions. This communication concerns with another interesting quadratic equation
PATTERN 1:
with four variables represented by X2 + Y2 = 2Z2 - 62W2. In (3), v 2 32u 2 Z 2
To start with, we observe that the following non-zero quadruples ( 2rs-1, 2rs-1, r2+s2, r2 -s2 ), (r2-s2 -1, r2-s2-
The solutions of the above equation is of the form
1,2rs, r2+s2), (y+2, y, y+2, ±1), (-1, y, -1, ±1), (-1, y, ±1, © 2017, IRJET
|
Impact Factor value: 5.181
|
ISO 9001:2008 Certified Journal
|
Page 1026