International Journal of Computational Engineering Research Vol, 0 Issue, Integral Solutions of the Non Homogeneous Ternary Quintic Equation ax by a b z a b ( ),, 0. S.Vidhyalakshmi 1, K.Lakshmi, M,A.Gopalan 1. Professor, Department of Mathematics,SIGC,Trichy. Lecturer, Department of Mathematics,SIGC,Trichy,. Professor, Department of Mathematics,SIGC,Trichy, Abstract: We obtain infinitely many non-zero integer triples ( x, y, z ) satisfying the ternary quintic ax by ( a b) z, a, b 0.Various interesting relations between the solutions and equation. special numbers, namely, polygonal numbers, Pyramidal numbers, Star numbers, Stella Octangular numbers, Pronic numbers, Octahedral numbers, Four Dimensional Figurative numbers and Five Dimensional Figurative numbers are exhibited. Keywords: Ternary quintic equation, integral solutions, -dimentional, -dimentional, - dimensional and - dimensional figurative numbers. MSC 000 Mathematics subject classification: 11D1. NOTATIONS: ( n1)( m) Tmn, n1 -Polygonal number of rank n with size m m 1 Pn n ( n 1)(( m ) n m ) - Pyramidal number of rank n with size m 6 SOn n(n 1) -Stella octangular number ofrank n Sn 6 n( n 1) 1-Star number of rank n PR n( n 1) -Pronic number of rank n n 1 OHn n(n 1) - Octahedral number of rank n n( n 1)( n )( n )( n ) F, n, = Five Dimensional Figurative number of rank n! whose generating polygon is a triangle. n( n 1)( n )( n ) F, n, = Four Dimensional Figurative number of rank n! whose generating polygon is a triangle 1. INTRODUCTION The theory of diophantine equations offers a rich variety of fascinating problems. In particular,quintic equations, homogeneous and non-homogeneous have aroused the interest of numerous mathematicians since antiquity[1-]. For illustration, one may refer [-10] for homogeneous and non-homogeneous quintic equations with three, four and five unknowns. This paper concerns with the problem of determining non-trivial integral solution of the non- homogeneous ternary quintic equation given by relations between the solutions and the special numbers are presented. ax by a b z a b ( ),, 0. A few www.ijceronline.com April 01 Page
II. METHOD OF ANALYSIS The Diophantine equation representing the quintic equation with three unknowns under consideration is ax by ( a b) z, a, b 0 (1) Introduction of the transformations x X bt, y X at () in (1) leads to X abt z () The above equation () is solved through different approaches and thus, one obtains distinct sets of solutions to (1).1. Case1: Choose a and b such that ab is not a perfect square. z p abq () Assume Substituting () in () and using the method of factorisation, define ( X abt ) ( p abq) () Equating real and imaginary parts in () we get, X p 10abp q ( ab) pq T qp 10 abp q ( ab) q In view of () and (), the corresponding values of xy, are represented by x p abp q ab pq b qp abp q ab q 10 ( ) ( 10 ( ) ) (7) y p abp q ab pq a qp abp q ab q 10 ( ) ( 10 ( ) ) (8) Thus (7) and () represent the non-zero distinct integral solutions to (1) A few interesting relations between the solutions of (1) are exhibited below: 1. a( x( p,1) by( p,1) 8 T, p1 Pp (1 10 ab) SOp p(mod10) ab. The following are nasty numbers; a( x( p,1) by( p,1) (i) 60 p 8 T, p1 Pp (1 10 ab) SOp (9 10 ab) p ab ax( p,1) by( p,1) ( ii)0 p 10F 0 F (ab )0P 1PR ab 1T 8T 60abT x( p,1) y( p,1). ab ( T, p ab) b a is a cubical integer., p,, p, p p, p, p, p. ax( p,1) by( p,1) ( a b){80f, p, 190F, p, 100Pp 70T, p 8T, T, } 0(mod) p p a( x( p,1) by( p,1) 8T, p1 Pp (1 10 ab) OH p 11( T, T, ) 0(mod10) p P ab 6. Sp 6 z( p,1) 18( OH p ) 6SOp 1T, p 6T, p 6ab 1. Now, rewrite () as, X abt 1 z (9) Also 1 can be written as ( a b ab)( a b ab) 1 ( ab) Substituting () and (10) in (9) and using the method of factorisation, define, www.ijceronline.com April 01 Page 6 (6) (10)
( a b ab) ( X abt ) ( p abq) (11) ab Equating real and imaginary parts in (11) we obtain, 1 X ( a b)( p 10abp q ( ab) pq ) ab(qp 10 abq p ( ab) q ) a b 1 T ( a b)(qp 10 abq p ( ab) q ) ( p 10abp q ( ab) pq ) a b In view of () and (), the corresponding values of xy, and z are obtained as, x ( a b) [( a b)( p 10abp q ( ab) pq ) b( a b)(qp 10 abp q ( ab) q )] y a b a b p abp q ab pq ( ) [( )( 10 ( ) ) a( a b)(qp 10 abp q ( ab) q )] z ( a b) ( p abq ) Further 1 can also be taken as ( ab ab)( ab ab) 1 ( ab ) For this choice, after performing some algebra the value of, x ab ab b p abp q ab pq xy and z are given by ( ) [( )( 10 ( ) ) ( ab b( ab ))(qp 10 abp q ( ab) q )] y ab ab a p abp q ab pq ( ) [( )( 10 ( ) ) z ( ab ) ( p abq ) Remark1: ( ab a( ab ))(qp 10 abp q ( ab) q )] Instead of (), the introduction of the transformations,, similar procedure we obtain different integral solutions to (1). (1) (1) (1) x X bt y X at in (1) leads to ().By a.. Case: Choose a and b such that ab is a perfect square,say, d. () is written as X ( dt ) z (1) To solve (1), we write it as a system of double equations which are solved in integers for XT, and in view of (), the corresponding integral values of xy, are obtained. The above process is illustrated in the following table: System of double equations X dt z X dt z X dt z X dt z Integral values of z z dk z dk Integral values of xy, x 8 k d ( d b) k( d b) y 8 k d ( d a) k( d a) x 8 k d ( d b) k( d b) y 8 k d ( d a) k( d a) www.ijceronline.com April 01 Page 7
X dt z X dt z X dt z X dt z X dt z X dt 1 X dt z X dt 1 z dk x k d ( d b) dk ( d b) y k d ( d a) dk ( d a) z dk x k d ( d b) dk ( d b) y k d ( d a) dk ( d a) z dk 1 x (16k d 0k d 0k d 0k d k)( d b) 1 y (16k d 0k d 0k d 0k d k)( d a) 1 z dk 1 x (16k d 0k d 0k d 0k d k)( d b) 1 y (16k d 0k d 0k d 0k d k)( d a) 1.. Case: Take z (16) Substituting (16) in (1) we get, X ( dt) ( ) (17) which is in the form of Pythagorean equation and is satisfied by rs, dt r s, X r s, r s 0 (18) Choose r and s such that rs 16d k (19) and thus dk. Knowing the values of rs, and using (18) and (), the corresponding solutions of (1) are obtained. For illustration, take r d k s dk r s,, 0 The corresponding solutions are given by 7 8 x 6 d k ( d b) dk ( d b) 7 8 y 6 d k ( d a) dk ( d a) (0) z d k Taking the values of rsdifferently, such that rs 16d k, we get different solution patterns. It is to be noted that, the solutions of (17) may also be written as X r s dt rs r s r s,,, 0 (1) Choose r, s Substituting the values of rs, in (1) and performing some algebra using () we get the corresponding solutions as 6 x d k ( d b) 8 d k ( d b) 6 y d k ( d a) 8 d k ( d a) () z d k Taking the values of rsdifferently, such that.. Case: Also,introducing the linear transformations X dk u, dt dk u in (1) and performing a few algebra, we have r s,we get different solution patterns. www.ijceronline.com April 01 Page 8
X k d dk, T k k d () In view of () and () the corresponding integral solutions of (1) are obtained as x k d ( d b) k( d b) () y k d ( d a) k( d a) z dk It is to be noted that, in addition to the above choices for X and dt, we have other choices, which are illustrated below: Illustration1: The assumption, in (1) yields X zx dt zdt () X md m n T nd m n z d m n ( ), ( ), ( ) (6) From (), (6) and (), the corresponding integral solutions of (1) are x d ( m n ) ( md nb) y d ( m n ) ( md na) z d ( m n ) Illustration: The assumption in (1) yields to X z X, dt z dt www.ijceronline.com April 01 Page 9 (6) (7) X ( dt) z (8) Taking z m n d (9) and performing some algebra, we get 10 9 X 16 m n d ( m n ), T 16 m n d ( n m ) (0) From (9), (0) and (), the corresponding integral solutions of (1) are 9 x 16 m n d [ d( m n ) b( n m )] 9 (1) y 16 m n d [ d( m n ) a( n m )] z m n d Illustration: Instead of (9), taking z dk 1 () in (8) we get X (dk 1) ( dk 1), T k(k 1) () From (0), (1) and (), the corresponding integral solutions of (1) are x (dk 1) ( dk bk 1) () y (dk 1) ( dk ak 1) z dk 1 Illustration: In (8), taking z t and arranging we get () X t ( dt ) (6) which is in the form of Pythagorean equation and is satisfied by X pq dt p q t p q r s,,, 0 (7) From (), (7) and () we get the corresponding solutions as 9 x ( p q ) d [ pqd b( p q )] 9 y ( p q ) d [ pqd a( p q )] z ( p q ) d It is to be noted that, the solutions of (6) may also be written as (8)
X p q t pq dt p q p q,,, 0 (9) From (),(7) and () we get the corresponding solutions as 9 x 16 p q d [ d( p q ) b( p q )] 9 y 16 p q d [ d( p q ) a( p q )] z p q d III. Remarkable observations: I: If ( x0, y0 z 0) be any given integral solution of (1), then the general solution pattern is presented in the, matrix form as follows: Odd ordered solutions: 6m 6m ( a b)( a b) a( a b) 0 x m1 x0 m 6m 6m ym 1 ( a b) a( a b) ( ab)( ab) 0 y 0 z m 1 0 0 1 z 0 Even ordered solutions: 6m 6m x ( ab)( ab) b( a b) 0 m x0 m 6m 6m y m ( a b) a( a b) ( ab)( ab) 0 y 0 z m 0 0 1 z 0 II: The integral solutions of (1) can also be represented as, x ( m bn) Z y ( m an) Z z m abn, a, b 0 IV. CONCLUSION In conclusion, one may search for different patterns of solutions to (1) and their corresponding properties. REFERENCES: [1]. L.E.Dickson, History of Theory of Numbers, Vol.11, Chelsea Publishing company, New York (19). []. L.Mordell, Diophantine equations, Academic Press, London (1969). []. Carmichael,R.D.,The theory of numbers and Diophantine Analysis,Dover Publications, New York (199) []. M.A Gopalan & A.Vijayashankar, An Interesting Diophantine problem Advances in Mathematics, Scientific Developments and Engineering Application, Narosa Publishing House, Pp 1-6, 010. (0) x y z []. M.A.Gopalan & A.Vijayashankar, Integral solutions of ternary quintic Diophantine equation x (k 1) y z,international Journal of Mathematical Sciences 19(1-), 16-169, (jan-june 010) [6]. M.A..Gopalan,G.Sumathi & S.Vidhyalakshmi, Integral solutions of non-homogeneous ternary quintic equation in terms of pells sequence x y xy( x y) Z, accepted for Publication in JAMS(Research India Publication) [7]. S.Vidhyalakshmi, K.Lakshmi and M.A.Gopalan, Observations on the homogeneous quintic equation with four unknowns x y z ( x y)( x y ) w,accepted for Publication in International Journal of Multidisciplinary Research Academy. (IJMRA) [8]. M.A.Gopalan &. A.Vijayashankar, Integral solutions of non-homogeneous quintic equation with five unknowns xy zw R, Bessel J.Math, 1(1),-0,011. [9]. M.A.Gopalan & A.Vijayashankar, solutions of quintic equation with five unknowns x y ( z w ) P,Accepted for Publication in International Review of Pure and Applied Mathematics. [10] M.A.Gopalan,G.Sumathi & S.Vidhyalakshmi, On the non-homogenous quintic equation with five unknowns x y z w 6T,accepted for Publication in International Journal of Multidisciplinary Research Academy (IJMRA). www.ijceronline.com April 01 Page 0