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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

[email protected]

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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[Pandichelvi,2(2): Feb., 2013] ISSN: 2277-9655

http: // www.ijesrt.com (C)International Journal of Engineering Sciences & Research Technology[306-311]

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.

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