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TRANSCRIPT
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[Pandichelvi,2(2): Feb., 2013]
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IJESRTINTERNATIONAL JOU
AN EXCLUS3
Department of M
The transcendental equation3 2
X
solutions. A few interesting relation
pyramidal numbers are presented.
Keywords: Transcendental equation, i
AMS Classification Number:11D 99
IntroductionDiophantine equations are numerously
require integral solutions are algebraic
equations are studied for its non-trivial
represented by3 23 22ZYX ++
interesting relations between the soluti
presented.
Notations
Polygonal
numbers
Gnomonic
Hexagonal
Stella Octang
Octahedral
Rhombic dodeca
Pentagonal pyra
Pronic
Centered Squ
Triangular
IS
ational Journal of Engineering Sciences & Research T
NAL OF ENGINEERING SCIENCES & R
TECHNOLOGY
IVE TRANSCENDENTAL EQUATION223 2222 )1( RkWZYX +=+++
V.Pandichelvi
thematics, Cauvery College for women, Trichy, India
Abstract
223 222 )1( RkWZY +=++ is analyzed for it
s between the solutions ),,,( WZYX , special poly
tegral solutions,polygonal numbers and pyramidal num
rich because of its variety. In [1-7], the Diophantine equ
quations with integer co-efficient. In [8-13], six differen
integral solutions. In this communication, the transcende
222 )1( RkW += is analyzed for its non trivial-inte
ns ),,,( WZYX , special polygonal numbers and pyram
Notations
for rank n Defi
nGno 2
nHex 2(n
ulanSO 2(n
nOH 2(
3
1n
gonalnRD 2)(12( n
midal
nPP 21 2n
noPr (n
renCS 2 2n
nT (n
SN: 2277-9655
echnology[306-311]
SEARCH
.
non trivial-integral
onal numbers and
ers.
tions for which we
t transcendental
tal equation
ral solutions. A few
idal numbers are
itions
1
)1
)12
)12 +n
)122
+ n
)1( +n
)1+
12 +n
2
)1+
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TetrahedralnTH )2)(1(
6
1++ nnn
Centered m-gonalnmCT ,
2
]2)1([ ++nnm
Decagonal
n
Dec )34( nn
Explicit formulas for the above m-gonal numbers may be found in [14-16].
Method of AnalysisThe transcendental equation to be solved is
223 223 22 )1( RkWZYX +=+++ (1) where
{ }0k
Taking
223 YX += (2)
The values of X and Ysatisfying (2) are offered by)( 22 nmmX += (3)
)( 22 nmnY += (4)
Similarly, by choosing
223 WZ +=
the values of Zand Wsatisfying the above cubic equation are stated by
)3( 22 nmmZ = (5)
)3( 22 mnnW = (6)
On substituting (3),(4),(5) and (6) in (1),we obtain
2222 )1()(2 Rknm +=+ (7)Set
22 baR +=
Applying unique factorization method, (7) can be written as
)2)(2)()(())()(1)(1( 2222 abibaabibaikiknimnimii ++=++
Thus,
)2)(())(1( 22 abibaiknimi ++=++
Equating real and imaginary parts, we search out
)2( 22 kabbaknm +=+
)2( 22 kabbaknm =
Solving the above two equations, we grasp
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[ ]abbaabbakm 2)2(2
1 2222++= (8)
[ ]abbabaabkn 2)2(2
1 2222+++= (9)
Here, the non-trivial integral solutions to (1) are analyzed when kis odd and kis even.
Case (i)
Consider 12 += k
This choice leads (8) and (9) to
2222 )2( baabbam ++= (10)
abbaabn 2)2( 22 ++= (11)
Substituting (10) and (11) in (3),(4),(5) and (6), the non-zero integral solutions to (1) are symbolized by
)122()()2( 22222222 +++++= babaabbaX
)122()(2)2( 222222 +++++= baabbaabY
+++= )61520156(2 65423324563 babbabababaaZ
++++ )61520156(3 6542332456 babbabababaa
++ )3103(12 53352 abbaba )1515( 642246 bbabaa +
++++= )61520156(2 65423324563 babbabababaaW
+++ )61520156(3 6542332456 babbabababaa
++ )1515(6 6422462 bbabaa )6206( 5335 abbaba +
To find the relations among the solutions, we consider the choice ba =
Hence,
)22(8 236 ++= aX
)122)(1(8 26 +++= aY
)362(8 236 ++= aZ
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)132(8 36 = aW
A few interesting relations among the solutions are expressed below:
1. ,468
CT
a
XY=
2. )6(mod010128 6
+ DecTHa
X
3. 188
26 ++=
+
TPPa
YX
4. )4(8 16 ++=+ GnoTaWX
5.
0)(8 1
6=++
+ GnoSOaW
6. 08)62(8 626 =++ aTPPaZ 7. 1616 +=+++ GnoaWZYX 8. Each of the following expressions provides a nasty number
(i) )Pr162(3 16
+++ oaWZX
(ii) )24(36
OHaX
(iii) )]4(8)[1(36
CSPPaY ++
Case (ii)
Choosing 2=k in (8) and (9), it reduces to
++=
2
2)2(
2222 abba
abbam
+++=
22)2(
2222 abbabaabn
Since our interest centers on finding integral solution, we observe that m and n are integers for Aa 2= and
Bb 2=
Thus,
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)2(2)2(4 2222 ABBAABBAm ++=
)2(2)2(4 2222 ABBABAABn +++=
In view of (3),(4),(5) and (5),the non-zero integer solutions fulfilling (1) are exhibited by
)14)(()2(2)2(48 2222222 ++++= BAABBAABBAX
)14)(()2(2)2(48 2222222 +++++= BAABBABAABY
++= ABBAABBAZ 2(2)2(4 2222
+++22222222 )2)(112(4)2)(34(4 ABBAABBA
22222 4)(64 BABA
+++= ABBABAABW 2(2)2(4 2222
{ ++ 22222222 )2)(112(4)2)(34(4 ABBAABBA 22222 4)(64 BABA
To obtain some properties between the solutions we select BA =
Thus,
)1248(128 236 += AX
)1248(128 236 +++= AY
)184)(12(128 26 ++= AZ
)184)(12(128 26 ++= AW
A few relations among the solutions are furnished below:
1. GnoTRDAX += 4(1286
2. 1116
4(128 ++ += GnoTRDAY
3. )2(128 126
++= GnoPPAY
4. 010246
=++ HexAZX
5. 116
10241024 ++ =+ GnoHexAWY
6. Each of the following expressions characterizes a nasty number
(i) )13(128 26
GnoOHAYX ++
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(ii)
GnoT
ZX
)18(6
+
+
References[1] Bhatia, B.L. and Supriya Mohanty.,Nasty numbers and their Characterizations, MathematicalEducation,
July- September 1985, 34-37.
[2] Dickson, L.E.,History of the theory numbers, Diophantine Analysis, Vol.2, New York: Dover, 2005.[3] Lang,S,.Algebraic Number Theory, Second ed.NewYork:Chelsea,1999.[4] Mordell,L.J., Diophantine equations, Academic Press, New York 1969.[5] Nagell,T.,Introduction to Number theory, Chelsea Publishing Company, New York, 1981.[6] Oistein Ore.,Number theory and its History, New York : Dover,1988.[7] Weyl,H.,Algebraic theory of numbers,Princeton,NJ:Princeton UniversityPress,1998.[8] Gopalan.M.A. and Devibala.S,A Remarkable transcendental equation, Antartica J.Math , 3(2)(2006),209-
215.
[9] Gopalan.M.A and Pandichelvi.V,A Special Transcendental equation 33 yxyxz ++= ,Pureand Applied Mathematical Sciences. (Accepted for Publications) (2008).
[10]Gopalan.M.A. and Pandichelvi.V.,On transcendental equation 33 yBxyBxz ++= ,Antartica J.Maths,6(1)(2009),55-58.
[11]Gopalan.M.A. ,Shanmuganantham.P and S.Sriram.,On transcendental equationyBxyBxz ++= ,Antartica J.Maths,7(5)(2010),509-515.
[12]Gopalan.M.A. and Kaligarani.J.,On transcendental equation zzyyxx +=+++ ,Diophantus Journal of Mathematics,1(1)(2012),9-14.
[13]Gopalan.M.A and Pandichelvi.V,Observations on the Transcendental Equation 3 22 ykxxz ++= ,Diophantus Journal of Mathematics,1(2)(2012),59-68.
[14]David Wells.,The Penguin Dictionary of Curious and interesting numbers, Penguin Books,1997.[15]John,H.Conway and Richard K.Guy,The Book of Numbers, Springer Verlag,NewYork, 1995.[16]Kapur, J.N.,Ramanujans Miracles, Mathematical Sciences Trust Society,1997.