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1. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 03 | Mar -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1026
OBSERVATIONS ON X2 + Y2 = 2Z2 - 62W2
G.Janaki* and R.Radha*
*Department of Mathematics, Cauvery college for women, Trichy
---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - The Quadratic equation with four unknowns
of the form X2 + Y2 = 2Z2 - 62W2 has been studied for its
non-trivial distinct integral solutions. A few interesting
relations among the solutions and special polygonal
numbers are presented.
Key Words: Quadratic equation with four unknowns,
Integral solutions.
NOTATIONS:
3,
7,
10,
12,
13,
( 1)
Triangular number of rank n.
2
(5 3)
Heptagonal number of rank n.
2
(4 3) Decagonal number of rank n.
(5 4) Dodecagonal number of rank n.
(11 9)
Tridecagonal numb
2
n
n
n
n
n
n n
T
n n
T
T n n
T n n
n n
T

 

 
  
  

 
17,
18,
er of rank n.
(15 13)
Heptadecagonal number of rank n.
2
(8 7) Octadecagonal number of rank n.
(2 1) Gnomonic number of rank n.
n
n
n
n n
T
T n n
Gno n

 
  
  
INTRODUCTION:
In [ 1 to 12] some special types of quadratic
equations with four unknowns have been analyzed for
theirnon-trivialintegralsolutions.Thiscommunication
concerns with another interesting quadratic equation
with four variables represented by X2 + Y2 = 2Z2 -62W2.
To start with, we observe that the following non-zero
quadruples ( 2rs-1, 2rs-1, r2+s2, r2 -s2 ), (r2-s2 -1, r2-s2-
1,2rs, r2+s2), (y+2, y, y+2, ±1), (-1, y, -1, ±1), (-1, y, ±1, -
1), (-y, y, ±1, -y) satisfy the equation under
consideration. In the above quadruples,anytwovalues
of the unknowns are the same. In [13], the Quadratic
equation with four unknowns of the form XY+X+Y+1=
Z2 -W2 has been studied for its non-trivial distinct
integral solutions.Thus,towardsthisendwesearchfor
non-zero distinct integral solutions of the equation
under consideration.Afewinterestingrelationsamong
the solutions are presented.
METHOD OF ANALYSIS:
The Quadratic Diophantine equation with four
unknowns under consideration is
X2 + Y2 = 2Z2 - 62W2 (1)
The substitution of the linear transformations
X= u + v, Y = v - u and W = u (2)
in (1) leads to
Z 2= 32 u2+v2 (3)
Four differentchoices of solutions to (3) are presented
below. Once the values of u and v are known ,using(2),
the corresponding values of X and Y are obtained.
PATTERN 1:
In (3), 2 2 2
32v u Z 
The solutions of the above equation is of the form

2. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 03 | Mar -2017 www.irjet.net p-ISSN: 2395-0072
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2 2
2 2
32
2
32
Z A B
u AB
v A B
 

 
(4)
Substituting the values of u and v in (2) , we get
2 2
2 2
2 2
( , ) 32 2
( , ) 2 32
( , ) 32
( , ) 2
X X A B A AB B
Y Y A B AB A B
Z Z A B A B
W W A B AB
   
   
  
 
(5)
Thus (5) gives the distinct integral solution of (1)
OBSERVATIONS:
1.    2
17, 13,1,8 ,1 34 2 6 19( )n n nW n Y n T T Gno     
2. 10,( ,2 1) ( ,2 1) 2 1A AX A A Y A A T Gno     
PATTERN 2:
(3) can be written as
Z 2 .1 = 32 u2+v2
Assuming Z = 32p2+q2 and write
1 =
( 32 2)( 32 2)
36
i i  
 
 
(6)
Substituting (6) in (3), using the method of
factorization we get
      
  
2 2
32 32 32 2 32 2
32 32
36
a ib ib i i
u iv u iv
   
  
Now define,
   
2
32 32 2
( 32 )
6
a ib i
u iv
 
  (7)
   
2
32 32 2
( 32 )
6
a ib i
u iv
 
 
Equating the real and imaginary parts in (7), we have
 
 
2 2
2 2
1
32 4
6
1
64 64 2
6
u a ab b
v a ab b
  
  
Since our interest is on finding integer solutions we have
choose
a and b suitably so that u and v are integers.
2 2
2 2
32 4
64 64 2
u A AB B
v A AB B
  
  
(8)
Thus, using the values of u and v and performing
a few calculations the values of ,X Y and Z are obtained
as follows:
2 2
2 2
2 2
2 2
( , ) 96 60 3
( , ) 32 68
( , ) 32 4
( , ) 192 6
X X A B A AB B
Y Y A B A AB B
W W A B A AB B
Z Z A B A B
   
   
   
  
(9)
Thus (9) represents the non-trivial solution integral
solution of (1)
OBSERVATIONS:
1.     18,,1 ,1 8 24( ) 22A AX A Y A T Gno   
2. 12, 7,(1, ) (1,2 1) 2 11 157n nZ n W n T T n     
PATTERN 3:
(3) can be written as
Z 2 - 32 u2 = v2 (10)
Assuming Z = a2- 32b2 and

3. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
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1 =
(6 32)(6 32)
4
  
 
 
(11)
Substituting (11) in (10) , using the method of
factorization
      
  
2 2
32 32 6 32 6 32
32 32
4
a b a b
Z u Z u
   
  
Now define,
   
   
2
2
32 6 32
( 32 )
2
32 6 32
( 32 )
2
a b
Z u
a b
Z u
 
 
 
 
(12)
Equating the like terms in (12), we get
 
 
2 2
2 2
1
6 64 192
2
1
12 32
2
Z a ab b
u a ab b
  
  
Since our interest is on finding integer solutions we have
choose a and b suitably so that u and Z are integers.
2 2
2 2
2 2
6 64 192
12 32
2 64
Z A AB B
u A AB B
v A B
  
  
 
(13)
Thus, using the values of u and v and performing a few
calculations the values of ,X Y and Z are obtained as
follows:
2 2
2 2
2 2
2 2
( , ) 3 12 32
( , ) 12 96
( , ) 12 32
( , ) 6 64 192
X X A B A AB B
Y Y A B A AB B
W W A B A AB B
Z Z A B A AB B
   
   
   
   
(14)
Thus (14) represents the non-trivial solution integral
solution of (1)
OBSERVATIONS:
1.     3,2 1,1 2 1,1 95 16 AW A Y A T    
2. When B=1, X+Y+Z+192 is a Nasty number.
PATTERN 4:
(3) can be written as
  
  
2 2 2
2 2 2
2
32
32
32
(32 )( )
32
( ) 32 (15)
( ) (16)
u v Z
u Z v
u Z v Z v
u u Z v Z v
Z v u A
u Z v B
Z v B uA
Z v A uB
 
 
  
  

 

 
 
Solving the equations (15) and (16) we get the
solutions
2 2
2 2
32
32
2
Z B A
v A B
u AB
  
  
 
Thus, using the values of u and v and performing a few
calculations the values of ,X Y and Z are obtained as
follows:
2 2
2 2
2 2
32 2
32 2 (17)
2
32
X A AB B
Y A AB B
W AB
Z A B
   
   
 
  
Thus (17) represents the non-trivial solution integral
solution of (1).
OBSERVATIONS:
1. ( , ) ( , ) ( , )Y A A X A A W A A  is a nasty number.

4. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 03 | Mar -2017 www.irjet.net p-ISSN: 2395-0072
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2. ( , ) ( , )Y A A X A A is a square number.
3. ( ,1) ( ,1) 2 ( ,1)AX A W A Gno Z A  
4. 2
( ,1) (2 , ) ( ,1) 6 2AY A W A A Z A O   
REFERENCE :
[1] Dickson.L.E., “History of the Theory of
Numbers”, Chelsea Publishing Company, New
York, Vol II, 1952.
[2] M.A.Gopalan, Manju Somanath and N. Vanitha,
“one Quadratic Diophantine equation with four
variables”, Bulletin of pure applied Sciences,
Vol.24E, No.2, pp.389-391 (2005).
[3] M.A. Gopalan, Manju Somanath and N. Vanitha,
“Integral Solutions of ax2 + by2 = w2 – z2”,
Advances in Theoretical and Applied
Mathematics, Vol.l, No.3.pp.223-226(2006).
[4] M.A. Gopalan and S. Vidyalakshmi, “Integral
solutions of x2+y2 =w2+Dz2”, Advances in
Theoretical and Applied Mathematics, Vol.l,
No.2,pp.115-118(2006).
[5] M.A. Gopalan and S. Vidyalakshmi, “Integral
solutions of kxy – w(x+y)=z2”,Advances in
Theoretical and Applied Mathematics,
Vol.1No.2,pp.167-172(2006)
[6] Mordell.L.J., Diophantine equation, Academic
Press, New York (1969)
[7] M.A. Gopalan and S. Vidyalakshmi, “On the
Diophantine equation kxy +yz+zx=w2”Pureand
Applied Mathematika Sciences, Vol. LXVIII, No.
1-2, September 2008.
[8] M.A. Gopalan, Manju Somanath and N. Vanitha,
“On a Quadratic Diophantine equationwithfour
variables”,AntarticaJ.math.,4(1),41-45,(2007).
[9] M.A. Gopalan, Manju Somanath and N. Vanitha,
“Homogeneous Quadratic equation with four
unknowns”, Acta ciencia Indica, Vol.XXXIIIM,
No.3, 915(2007).
[10] M.A. Gopalan, Manju Somanath and N. Vanitha,
“On Space Pythagorean equationx2+y2+z2 =w2”,
International journalofMathematics,Computer
Sciences and Information Technology,
Vol.1,No.1, January-June, pp.129-1332008.
[11] M.A. Gopalan and S.Vidyalakshmi, “Quadratic
Diophantine equation with four variables
x2+y2+xy+y=u2+v2uv+u-v”. Impact J.Sci. Tech:
Vol 2(3), 125-127, 2008.
[12] M.A. Gopalan, Manju Somanath andN.Vanitha,“
On Quadratic Diophantine equation with four
unknowns x2+y2=z2+2w2-2w+1 Impact J.Sci.
Tech: Vol 2(2), 97-103, 2008.
[13] G.Janaki andS.Vidyalakshmi,“Integralsolutions
of xy+x+y+1 = z2 - w2 ”, Antartica J.math., 7(1),
31-37, (2010).

5. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 03 | Mar -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1030
BIOGRAPHIES
Dr.G.JanakireceivedthePh.D.,
M.Sc., and M.Phil., degree in
mathematics from
Bharathidasan University,
Trichy, South India. She
completed her Ph.D. degree
National College,
Bharathidasan University.
She has published many
papers in international and
national level journals. Her
research area is “Number
Theory”.
R.Radha received M.Sc.
degree in mathematics from
Bharathidasan University,
Trichy and M.Phil degree in
mathematics from Alagappa
University, Karaikudi, South
India. Her ongoing research
focusing on the subject of
“Number Theory”.