SPECIAL INTEGER QUADRUPLE IN ARITHMETIC PROGRESSION
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1 SPECIAL INTEGER QUADRUPLE IN ARITHMETIC PROGRESSION K.Meena 1, S.Vidhyalakshmi, M.A.Gopalan, S. Aarthy Thangam 4 1 Former VC, Bharathidasan University, Trichy-0 04, Tamil Nadu, India., Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-0 00, Tamil Nadu, India. 4 Assistant Professor, Department of Mathematics, Chidambaram Pillai College for Women, Mannachanallur, Trichy-0, Tamil Nadu, India. Abstract: We search for integer quadruples (x, y, z, w) in arithmetic progression such that x y = α, z β, x y z γ Keywords: Integer quadruple, quadruple in A.P, system of equations. I. INTRODUCTION Let a and d be two non-zero distinct integers. In [1], the authors have obtained triples in Arithmetic Progression (a-d, a, ad) such that 1) a d = α, a d = β, a = χ 4 ) a d = α, a d = β, a = χ In [], the authors have obtained triples in Arithmetic Progression (a-d, a, ad) such that each of the expressions a ad, ad and a represents a perfect square. In [], triples in Arithmetic Progression such that the sum of any two is a perfect square are obtained. In this communication, integer quadruples in Arithmetic Progression (A.P) satisfying certain relations among its members are obtained. II. METHOD OF ANALYSIS Let (x, y, z, w) be four non-zero distinct integers such that the quadruple (x, y, z, w) is in A.P. Let a, d be two non-zero integers. Then, we know that, the quadruple ( a d,a d,a d,a d) represents an arithmetic progression. Therefore we take x = a d, y = a d, z = a d, a d.the problem under consideration is to find a and d such that x y = a 4d = α (1) From (), we ve z x a 4d = β () y z 4a = γ () a = p (4) Solving (1) and () for a, d we ve α β β α a =, d = 4 8 Since a and d are to be integers, we choose α = 4P, β = 4Q a = 4P 4Q () d = Q P () DOI:.88/IJRTER DDV9E 8
2 Volume 0, Issue 0; May [ISSN: 4-147] From (4) and () we ve, ( P Q ) = p (7) The above equation is solved in two different ways. WAY : 1 Assume p = u v (8) Write as = (1 i)(1 i) (9) Substituting (8) and (9) in (7) and applying the method of factorization, define ( 1 i)( P iq) = ( u iv) P Q = u uv P Q = u v v 1 P = ( u u v uv v ) () 1 Q = ( u u v uv v ) (11) Replacing u by R and v by S in (8), (), (11) we ve p = 4R 4S P = 4R 1RS 1R S 4S Q = 4R 1RS 1R S 4S Substituting the above values of p, P, Q in (4) and () we ve 4 4 a = 18R 84R S 84R S 18S d = 84R S 180R S 84RS 4 4 x = 18R 11R S 84R S 840R S 84R S 11RS 18S to (). 4 4 y = 18R 84R S 84R S 180R S 84R S 84RS 4 4 z = 18R 84R S 84R S 180R S 84R S 84RS 18S 18S R 11R S 84R S 840R S 84R S 11RS 18S Note that the quadruple (x, y, z, w) is in A.P whose members satisfy the relations (1) Numerical Examples: m n x y z w x y z w x y z w Properties: 1. { y( R,S) z( R,S) } is a cubical integer. x R,S w R,S is a cubical integer.. { }. y( R,1) z( R,1) ( CP,R ) 78( T4,R ) 18SR 0( mod ) 4. z( S,S) x( S,S) y( All Rights Reserved 9
3 WAY : It is observed that P = m m n satisfy (7) From (4) we ve ( ) ( m n ) ( m n ) Q = n p = ( m n ) a = 1 From () we ve Now, International Journal of Recent Trends in Engineering & Research (IJRTER) Volume 0, Issue 0; May [ISSN: 4-147] ( n m )( m n ) d = 8 x = 8 ( m n ) ( m n ) ( m n ) ( m n ) ( m n ) ( m n ) 8( m n ) ( m n ) y = 8 z = 8 Note that the quadruple (x, y, z, w) is in A.P whose members satisfy the required conditions. Numerical Examples: m n x y z w x y z w x y z w Properties: x m,n w m,n is a cubical integer. 1. ( ). If n = rs, m = r s, then ( w) y is a perfect square.. F ( CP ) x( m,1) z( m,1) 40( OH ) 88T 0( mod ) 4 4,m,4,m m 4, m 4. x ( m,m) y( m,m) z( m,m). w( n,n) y( n,n) However, we have two more patterns of quadruples satisfying (1) to () which are illustrated as below: WAY : In addition to (9), may also be written as ( 1 7i) ( 1 7i) = (1) Substituting (8) and (1) in (7) and applying the method of factorization, define ( 1 7i) P iq = u iv P 7Q = u 1uv 7P Q = 1u v All Rights Reserved 1
4 Volume 0, Issue 0; May [ISSN: 4-147] 1 P = ( u 1u v uv 7v ) (1) 1 Q = ( 7u u v 1uv v ) (14) Replacing u by R and v by S in (8), (1), (14) we ve p = 0R 0S P = 0R 0R S 00RS 700S Q = 700R 00R S 0RS 0S Substituting the above values of p, P, Q in (4) and () we ve 4 4 a = 0 R R S R S S 4 4 d = ( 0) ( 9R 18R S 1440R S 0R S 1440R S 18RS 9S ) 4 4 ( 0) ( 88R 04R S 490R S 180R S 70R S 04RS 488S ) 4 4 ( 0) ( 4R 18R S 040R S 0R S 840R S 18RS 9S ) 4 4 ( 0) ( 9R 18R S 840R S 0R S 040R S 18RS 4S ) 4 4 ( 0) ( 488R 04R S 70R S 180R S 490R S 04RS 88S ) x = y = z = Note that the quadruple (x, y, z, w) is in A.P whose members satisfy the relations (1) to (). WAY: 4 In addition to (9), may also be written as ( 7 i) ( 7 i) = (1) Substituting (8) and (1) in (7) and applying the method of factorization, define ( 7 i) P iq = u iv 7P Q = u 1uv P 7Q = 1u v v 1 P = ( 7u u v 1uv v ) (1) 1 Q = ( u 1u v uv 7v ) (17) Replacing u by R and v by S in (8), (1), (17) we ve p = 0R 0S P = 700R 00R S 0RS 0S Q = 0R 0R S 00RS 700S Substituting the above values of p, P, Q in (4) and () we ve 4 4 a = 0 R R S R S S x = 4 4 d = ( 0) ( 9R 18R S 1440R S 0R S 1440R S 18RS 9S ) 4 4 ( 0) ( 488R 04R S 70R S 180R S 490R S 04RS 88S ) 4 4 ( 0) ( 9R 18R S 840R S 0R S 040R S 18RS 4S ) 4 4 ( 0) ( 4R 18R S 040R S 0R S 840R S 18RS 9S ) 4 4 ( 0) ( 88R 04R S 490R S 180R S 70R S 04RS 488S ) y = z All Rights Reserved 111
5 Volume 0, Issue 0; May [ISSN: 4-147] Note that the quadruple (x, y, z, w) is in A.P whose members satisfy the required conditions. Note: It is worth to note that one may obtain some more quadruples satisfying (1) to () by assuming as follows: = 1 i 1 i = ( 7 i) ( 7 i) ( 1 7i) ( 1 7i) = REFERENCES [1] M.A. Gopalan, V. Geetha, V. Krithika, On Two Special Integer Triples in Arithmetic Progression, Open Journal of Applied & Theoretical Mathematics (OJATM), Vol., No.1, March 01, pp [] S. Vidhyalakshmi, A. Kavitha, M.A. Gopalan, Diophantine Problem on Integer Triple is Arithmetic Progression, Transactions on Mathematics TM, Vol., No., April 01, pp -4. [] M.A. Gopalan, V. Sangeetha, An Interesting Diophantine Problem, Open Journal of Applied & Theoretical Mathematics (OJATM), Vol., No., June 01, pp All Rights Reserved 11
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