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Construction of Sequences of Diophantine 3-Tuples Through the Ratio of Pentatope and Tetrahedral Numbers

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Construction of Sequences of Diophantine 3-Tuples Through the Ratio of Pentatope and Tetrahedral Numbers AEGAEUM JOURNAL ISSN NO: 0776-3808 CONSTRUCTION OF SEQUENCES OF DIOPHANTINE 3-TUPLES THROUGH THE RATIO OF PENTATOPE AND TETRAHEDRAL NUMBERS A. Vijayasankar1, Sharadha Kumar2 & M.A. Gopalan3 1Assistant Professor, Department of Mathematics, National College, Affiliated to Bharathidasan University, Trichy-620 001, Tamil Nadu, India. email: [email protected] 2Research Scholar, Department of Mathematics, National College, Affiliated to Bharathidasan University ,Trichy-620 001, Tamil Nadu, India. email: [email protected] 3Professor, Department of Mathematics, Shrimati Indira Gandhi College, Affiliated to Bharathidasan University, Trichy-620 002, Tamil Nadu, India. email:[email protected] Abstract: This paper deals with the study of constructing sequences of diophantine triples a ,b ,c through the ratio of Pentatope and Tetrahedral Numbers such that the product of any two elements of the set added by a polynomial with integer coefficient is a perfect square. Keywords: Diophantine 3-tuples, Sequences of 3-tuples, Pentatope Numbers, Tetrahedral Numbers. 1. INTRODUCTION The problem of constructing the sets with property that product of any two of its distinct elements is one less than a square has a very long history and such sets have been studied by Diophantus. A set of m distinct positive integers a1 ,a 2 ,a 3 ,.....,a m is said to have the property Dn,n Z 0 if a ia j n is a perfect square for all 1 i j m or 1 j i m and such a set is called a Diophantine m-tuple with property Dn. Many Mathematicians considered the construction of different formulations of diophantine triples with the property Dn for any arbitrary integer n [1] and also, for any linear polynomials in n. In this context, one may refer [2-15] for an extensive review of various problems on diophantine triples. This paper aims at constructing sequences of diophantine triples through the ratio of Pentatope and Tetrahedral Numbers where the product of any two members of the triple added with the polynomial with integer coefficients satisfies the required property. 2. NOTATIONS nn 1n 2n 3 1. PT Pentatope Number of rank n n 24 nn 1n 2 2. P3 Tetrahedral Number of rank n n 6 Volume 8, Issue 3, 2020 http://aegaeum.com/ Page No: 1057 AEGAEUM JOURNAL ISSN NO: 0776-3808 3. METHOD OF ANALYSIS Sequence 1: PTnk3 PTnk3 Let a 4 3 n k , c0 4 3 n k Pnk3 Pnk3 It is observed that 2 2 ac0 k n 2 Therefore, the pair a ,c0 represents diophantine 2-tuple with the property D(k ) . Let c1 be any non-zero polynomial such that 2 2 ac1 k p (1) 2 2 c0c1 k q (2) Eliminating c1 between (1) and (2), we have 2 2 2 c0p aq c0 ak (3) Introducing the linear transformations p X aT , q X c0T (4) in (3) and simplifying we get 2 2 2 X ac0T k which is satisfied by T 1 , X n In view of (4) and (1), it is seen that c1 4n 2 Note that a ,c0 ,c1 represents diophantine 3-tuple with property D(k ) Taking a ,c1 and employing the above procedure, it is seen that the triple a ,c1 ,c2 where c 2 9n 3k 2 exhibits diophantine 3-tuple with property D(k ) Volume 8, Issue 3, 2020 http://aegaeum.com/ Page No: 1058 AEGAEUM JOURNAL ISSN NO: 0776-3808 Taking a ,c2 and employing the above procedure, it is seen that the triple a ,c2 ,c3 where c3 16n 8k exhibits diophantine 3-tuple with property D(k2 ) Taking a ,c3 and employing the above procedure, it is seen that the triple a ,c3 ,c4 where c 4 25n 15k 2 exhibits diophantine 3-tuple with property D(k ) The repetition of the above process leads to the generation of sequence of diophantine 3- tuples whose general form is given by a ,cs1 ,cs where 2 2 cs1 s n s 2sk , s 1,2,3,... A few numerical examples are presented in Table 1 below: Table 1: Numerical Examples n k D a,c ,c a,c ,c a ,c ,c a ,c ,c 0 1 1 2 2 3 3 4 2 2 4 (4, 0, 8) (4, 8, 24) (4, 24, 48) (4, 48, 80) 3 3 9 (6, 0, 12) (6, 12, 36) (6, 36, 72) (6, 72, 120) 4 4 16 (8, 0, 16) (8, 16, 48) (8, 48, 96) (8, 96, 160) 5 5 25 (10, 0, 20) (10, 20, 60) (10, 60, 120) (10, 120,200) Sequence 2: PTn1 PTn5 Let a 4 3 n 2, c0 4 3 n 2 Pn1 Pn5 It is observed that 2 ac0 4 n Therefore, the pair a ,c0 represents diophantine 2-tuple with the property D (4) . Let c1 be any non-zero polynomial such that 2 ac1 4 p (5) Volume 8, Issue 3, 2020 http://aegaeum.com/ Page No: 1059 AEGAEUM JOURNAL ISSN NO: 0776-3808 2 c0c1 4 q (6) Eliminating c1 between (5) and (6), we have 2 2 c0p aq c0 a4 (7) Introducing the linear transformations p X aT , q X c0T (8) in (7) and simplifying we get 2 2 X ac0T 4 which is satisfied by T 1 , X n In view of (8) and (5), it is seen that c1 4n Note that a ,c0 ,c1 represents diophantine 3-tuple with property D (4) Taking a ,c1 and employing the above procedure, it is seen that the triple a ,c1 ,c2 where c 2 9n 6 exhibits diophantine 3-tuple with property D (4) Taking a ,c2 and employing the above procedure, it is seen that the triple a ,c2 ,c3 where c3 16n 16 exhibits diophantine 3-tuple with property D (4) Taking a ,c3 and employing the above procedure, it is seen that the triple a ,c3 ,c4 where c 4 25n 30 exhibits diophantine 3-tuple with property D (4) The repetition of the above process leads to the generation of sequence of diophantine 3- tuples whose general form is given by a ,cs1 ,cs where Volume 8, Issue 3, 2020 http://aegaeum.com/ Page No: 1060 AEGAEUM JOURNAL ISSN NO: 0776-3808 2 2 cs1 s n 2s 4s , s 1,2,3,... A few numerical examples are presented in Table 2 below: Table: 2 Numerical Examples n D a,c ,c a,c ,c a ,c ,c a,c ,c 0 1 1 2 2 3 3 4 2 4 (4, 0, 8) (4, 8, 24) (4, 24, 48) (4, 48, 80) 3 4 (5, 1, 12) (5, 12, 33) (5, 33, 64) (5, 64, 105) 4 4 (6, 2, 16) (6, 16, 42) (6, 42, 80) (6, 80, 130) 5 4 (7, 3, 20) (7, 20, 51) (7, 51, 96) (7, 96, 155) Sequence 3: PTn PTn6 Let a 4 3 n 3, c0 4 3 n 3 Pn Pn6 It is observed that ab 9 n 2 Therefore, the pair a ,c0 represents diophantine 2-tuple with the property D (9) . Let c1 be any non-zero polynomial such that 2 ac1 9 p (9) 2 c0c1 9 q (10) Eliminating c1 between (9) and (10), we have 2 2 c0p aq c0 a9 (11) Introducing the linear transformations p X aT , q X c0T (12) in (11) and simplifying we get 2 2 X ac0T 9 which is satisfied by T 1 , X n In view of (12) and (9), it is seen that Volume 8, Issue 3, 2020 http://aegaeum.com/ Page No: 1061 AEGAEUM JOURNAL ISSN NO: 0776-3808 c1 4n Note that a ,c0 ,c1 represents diophantine 3-tuple with property D (9) Taking a,c1 and employing the above procedure, it is seen that the triple a,c1,c2 where c2 9n 9 exhibits diophantine 3-tuple with property D (9) Taking a ,c2 and employing the above procedure, it is seen that the triple a ,c2 ,c3 where c3 16n 24 exhibits diophantine 3-tuple with property D (9) Taking a ,c3 and employing the above procedure, it is seen that the triple a ,c3 ,c4 where c 4 25n 45 exhibits diophantine 3-tuple with property D (9) The repetition of the above process leads to the generation of sequence of diophantine 3- tuples whose general form is given by a ,cs1 ,cs where 2 2 cs1 s n 3s 6s, s 1,2,3,... A few numerical examples are presented in Table 3 below: Table 3: Numerical Examples n D a,c ,c a,c ,c a ,c ,c a,c ,c 0 1 1 2 2 3 3 4 2 9 (5, -1, 8) (5, 8, 27) (5, 27, 56) (5, 56, 95) 3 9 (6, 0, 12) (6, 12, 36) (6, 36, 72) (6, 72, 120) 4 9 (7, 1, 16) (7, 16, 45) (7, 45, 88) (7, 88, 145) 5 9 (8, 2, 20) (8, 20, 54) (8, 54, 104) (8, 104, 170) Volume 8, Issue 3, 2020 http://aegaeum.com/ Page No: 1062 AEGAEUM JOURNAL ISSN NO: 0776-3808 Sequence 4: PT PT Let n n2 a 4 3 n 3, c0 4 3 n 1 Pn Pn2 It is observed that 2 2 ac0 2k 2n k 2k 2 n k 1 Therefore, the pair a ,c0 represents diophantine 2-tuple with the property D2k 2n k2 2k 2 . Let c1 be any non-zero polynomial such that 2 2 ac1 2k 2n k 2k 2 p (13) 2 2 c0c1 2k 2n k 2k 2 q (14) Eliminating c1 between (13) and (14), we have 2 2 2 c0p aq c0 a2k 2n k 2k 2 (15) Introducing the linear transformations p X aT , q X c0T (16) in (15) and simplifying we get X2 abT2 2k 2n k 2 2k 2 which is satisfied by T 1 , X n k 1 In view of (16) and (13), it is seen that c1 4n 2k 6 Note that a ,c0 ,c1 represents diophantine 3-tuple with property D(2k 2n k2 2k 2) Taking a ,c1 and employing the above procedure, it is seen that the triple a , c1 ,c2 where c 2 9n 4k 17 2 exhibits diophantine 3-tuple with property D(2k 2n k 2k 2) Volume 8, Issue 3, 2020 http://aegaeum.com/ Page No: 1063 AEGAEUM JOURNAL ISSN NO: 0776-3808 Taking a ,c2 and employing the above procedure, it is seen that the triple a ,c2 ,c3 where c3 16n 6k 34 2 exhibits diophantine 3-tuple with property D(2k 2n k 2k 2) The repetition of the above process leads to the generation of sequence of diophantine 3- tuples whose general form is given by a ,cs1 ,cs where 2 2 cs1 s n 2s 1k 3s 4s 2 , s 1,2,3,..
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