1. The document analyzes the transcendental equation with five unknowns represented by 5222 224 22 )1( Rkwzyx n  . 2. Two methods of analysis are presented to obtain non-zero distinct integral solutions to the equation. The first method involves substitutions that represent the solutions in terms of polynomials involving A and B. 3. The second method uses different substitutions to represent the solutions in terms of polynomials involving m and n, providing a different pattern of solutions. Properties of the solutions are also outlined.

1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 4 (Jul. - Aug. 2013), PP 78-81
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www.iosrjournals.org 78 | Page
On The Surd Transcendental Equation With Five Unknowns
5222 224 22
)1( Rkwzyx n

M.A.Gopalan1
, G.Sumathi2
and S.Vidhyalakshmi3
1,2,3
Department of Mathematics, Shrimathi Indira Gandhi College,Trichy-600002,India.
Abstract: The transcendental equation with five unknowns represented by the diophantine equation
5222 224 22
)1( Rkwzyx n
 is analyzed for its patterns of non-zero distinct integral
solutions.
Keywords: Transcendental equation,integral solutions,surd equation
M.Sc 2000 mathematics subject classification: 11D99
NOTATIONS
nmt , : Polygonal number of rank n with size m
m
nP : Pyramidal number of rank n with size m
m
nCp : Centered Pyramidal number of rank n with size m
I. Introduction
Diophantine equations have an unlimited field of research by reason of their variety. Most of the
Diophantine problems are algebraic equations  3,2,1 .It seems that much work has not been done to obtain
integral solutions of transcendental equations.In this context one may refer  164  . This communication
analyses a transcendental equation with five unknowns given by
5222 224 22
)1( Rkwzyx n
 .
Infinitely many non-zero integer quintuples )w,z,Y,X,y,x( satisfying the above equation are obtained.
II. Method Of Analysis
The diophantine equation representing a transcendental equation with five unknowns is
5222 224 22
)1( Rkwzyx n
 (1)
To start with,the substitution of the transformations
................................................
)(4
)(4
22222
22






qpqpy
qppqx
(2a)
................................................................
2
22 





qpw
pqz
(2b)
in(1) leads to
5n2222
R)1k()qp(2  (3)
Assume 0B,A,BA)B,A(RR 22
 (4)
and write 2 as )i1)(i1(2  (5)
Substituting (5),(4) in (3),and employing the method of factorization,define
5
)iBA)(i()iqp)(i1(  (6)

2. On The Surd Transcendental Equation With Five Unknowns
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where ,
 
 nn
nn
ikik
i
ikik
22
22
)()(
2
1
)()(
2
1




Equating real and imaginary parts in (6),we get
.....................................
),(),(
),(),(





BAgBAfqp
BAgBAfqp


(7)
where
)BBA10BA5()B,A(g
)AB5BA10A()B,A(f
5324
4235


Solving the system of equations (7),we get
...............................
2
)B,A(g)()B,A(g)(
q
2
)B,A(g)()B,A(f)(
p










(7a)
It is to be noted that p and q are integers only when A and B are of the same pairty.
Replacing A by A2 , B by B2 in(7a) and (4) ,we have
 
 
......................
),()(),()(2
,()(),()(2
4
4






BAgBAgq
BAgBAfp


(7b)
)BA(4R 22
 (7c)
Substituting (7b) in (2a) and (2b), the values of )w,z,y,x( are represented by
  
      
   
      
  
   228
8
222
22
16
22
16
)B,A(f)()B,A(g)()B,A(g)()B,A(f)(2)B,A(w
)B,A(f)()B,A(g)()B,A(g)()B,A(f)(2)B,A(z
)B,A(f)()B,A(g)()B,A(g)()B,A(f)(
)B,A(f)()B,A(g)()B,A(g)()B,A(f)(4
2)B,A(y
)B,A(f)()B,A(g)()B,A(g)()B,A(f)(
)B,A(f)()B,A(g)()B,A(g)()B,A(f)(4
2)B,A(x
























The above equations and (7c) represents the non-zero integer solutions to (1).
In a similar manner , replacing A by 2A+1, B by 2B+1 one obtains the corresponding set of non-zero integer
solutions to (1).
It is worth to observe that one may employ different set of transformations for z and w leading to a different
solution pattern which is illustrated as follows.
in (1), we get
222
swz  (8)
(8) can be rewritten as
12222
 sswz (9)
Assume 0,,),( 22
 qpqpqpss (10)
and write 1 as

3. On The Surd Transcendental Equation With Five Unknowns
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................................
)(
)2)(2(
1
222
2222







nm
mninmmninm
(11)
Substituting (11),(10) in (9),and employing the method of factorization,define
 
 
...............................
)(
)(2)(2
)(
4))((
22
2222
22
2222













nm
nmpqqpmn
w
nm
pqmnqpnm
z
(12)
As our thurst is an finding integer solution,replacing p by Pnm )( 22
 , q by Qnm )( 22
 in
(2a),(12),we have
 
 
 
...........
)(2)(2)(
4))(()(
)(4)(
)()(4
222222
222222
22222422
22422












nmPQQPmnnmw
PQmnQPnmnmz
QPQPnmy
QPPQnmx
(13)
where
      
      )B,A(g)B,A(fmn2)nm()B,A(g)B,A(fmn2)nm()nm(2Q
)B,A(g)B,A(fmn2)nm()B,A(g)B,A(fmn2)nm()nm(2P
22228224
22228224


Note that (13) and (4) represent the integral solutions to (1),provided A and B are of the same pairty.
2.1 : Properties:
1. ),,( ywz satisfies the hyperbolic paraboloid ywz  22
2.
4
12),1(),1( qPqqzqqw 
3. wzx 2
4. 18),1( ,3
2
 qtqqw
5.   5
4),1( qPqqqz 
6.
3
112)1,()1,(  pPpwpz
7.   3,12)1,(6)1,()1,(  mCppzpwpzm m
p
8. 2,4)1,()2()1,(2 ,  mtppzmpz pm
9. zwy )( 2
 is a cubical integer
10.
242
)( zwxzyz 
11. )(4 222
wywx 
12. 12  yw is a difference of two squares.
III. Conclusion:
To conclude,one may search for other pattern of solutions and their corresponding properties.

4. On The Surd Transcendental Equation With Five Unknowns
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[2] L.J.Mordel, Diophantine Equations, Academic press,Newyork,1969.
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[1985]
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58,(2009).
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Journal of mathematical sciences,Vol.9, No.1-2, 177-182, Jan-Jun 2010.
[7] M.A.Gopalan and V.Pandichelvi, ‘‘Observations on the Transcendental equation
3 22 ykxxz  ’’ Diophantus
J.Math.,1(2), 59-68, (2012).
[8] M.A.Gopalan and J.Kaliga Rani, ‘‘On the Transcendental equation zzyyxx  ’’
Diophantus.J.Math.1(1),9-14, 2012.
[9] M.A.Gopalan,Manju Somanath and N.Vanitha, ‘‘On Special Transcendenta Equations’’Reflections des ERA-JMS, Vol.7,issue
2,187-192,2012.
[10] V.Pandichelvi, ‘‘An Exclusive Transcendental equations
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R)1k(wzyx  ’’ International
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[11] M.A.Gopalan, S.Vidhyalakshmi and S.Mallika , ‘‘ On the Ttranscendental
equation
5223 223 22
)(2 RskWZYX  ’’,IJMER,Vol.3(3), 1501-1503, 2013.
[12] M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, “Observations On
2n23 222 22
z)3k(YX2x2y  ’’
International Journal of Pure and Applied Mathematical Sciences,vol 6,No 4,pp.305-311,2013.
[13] M.A.Gopalan,G.Sumathi and S.Vidhyalakshmi, ‘‘ On the Transcendental equation with five unknowns
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z)sr(YX2yx3  ’’ Global Journal of Mathematics and Mathematical
Sciences,Vol.3,No.2,pp.63-66,2013.
[14] M.A.Gopalan,G.Sumathi and S.Vidhyalakshmi, , ‘‘ On the Transcendental equation with six unknowns
2 223 222 22
w2zYXxyyx2  ’’, accepted in Cayley Journal of Mathematics
[15] M.A.Gopalan, S.Vidhyalakshmi and S.Mallika , ‘‘An interesting Ttranscendental
equation
23 222 22
RWZ2X3Y6  ’’,accepted in Cayley Journal of Mathematics
[16] M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, ‘‘On the Ttranscendental equation
23 222 22
zYX2x2y7  ’’ accepted in Diophantus Journal of Mathematics.