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
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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 )
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Taking a ,c2 and employing the above procedure, it is seen that the triple a ,c2 ,c3 where
c3 16n 8k
2 exhibits diophantine 3-tuple with property D(k )
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)
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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
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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
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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)
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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)
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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,...
A few numerical examples are presented in Table 4 below:
Table 4: 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 10 (5, 3, 18) (5, 18, 43) (5, 43, 78) (5, 78,123 )
3 3 25 (6, 4, 24) (6, 24, 56) (6, 56, 100) (6, 100, 156)
4 4 46 (7, 5, 30) (7, 30, 69) (7, 69, 122) (7, 122, 189)
5 5 73 (8, 6, 36) (8, 36, 82) (8, 82, 144) (8, 144, 222)
Sequence 5:
PT PT Let n1 n2 a 4 3 n 2, c0 4 3 n 1 Pn1 Pn2
It is observed that
2 2 ac0 2k 3n k 2 n k
Therefore, the pair a,c0 represents diophantine 2-tuple with the property
D2k 3n k2 2 .
Let c1 be any non-zero polynomial such that
2 2 ac1 2k 3n k 2 p (17)
2 2 c0c1 2k 3n k 2 q (18)
Eliminating c1 between (17) and (18), we have
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2 2 2 c0p aq c0 a2k 3n k 2 (19)
Introducing the linear transformations
p X aT , q X c0T (20)
in (19) and simplifying we get
2 2 2 X ac0T 2k 3n k 2
which is satisfied by T 1 , X n k
In view of (20) and (17), it is seen that
c1 4n 2k 3
2 Note that a ,c0 ,c1 represents diophantine 3-tuple with property D(2k 3n k 2)
Taking a ,c1 and employing the above procedure, it is seen that the triple a , c1 ,c2 where
c 2 9n 4k 9
2 exhibits diophantine 3-tuple with property D(2k 3n k 2)
Taking a ,c2 and employing the above procedure, it is seen that the triple a ,c2 ,c3 where
c3 16n 6k 19
2 exhibits diophantine 3-tuple with property D(2k 3n k 2)
Taking a ,c3 and employing the above procedure, it is seen that the triple a ,c3 ,c4 where
c 4 25n 8k 33
2 exhibits diophantine 3-tuple with property D(2k 3n k 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 2s 4s 3 , s 1,2,3,...
A few numerical examples are presented in Table 5 below:
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Table 5: 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, 3, 15) (4, 15, 35) (4, 35, 63) (4, 63, 99)
3 3 16 (5, 4, 21) (5, 21, 48) (5, 48, 85) (5, 85, 132)
4 4 34 (6, 5, 27) (6, 27, 61) (6, 61, 107) (6, 107, 165)
5 5 58 (7, 6, 33) (7, 33, 74) (7, 74, 129) (7, 129, 198)
Sequence 6:
PT PT Let n n1 a 4 3 n 3, c0 4 3 n 2 Pn Pn1
It is observed that
2 ac0 n 3 n 3
Therefore, the pair a ,c0 represents diophantine 2-tuple with the property Dn 3 .
Let c1 be any non-zero polynomial such that
2 ac1 n 3 p \ (21)
2 c0c1 n 3 q (22)
Eliminating c1 between (21) and (22), we have
2 2 c0p aq c0 an 3 (23)
Introducing the linear transformations
p X aT , q X c0T (24)
in (23) and simplifying we get
2 2 X ac0T n 3
which is satisfied by T 1 , X n 3
In view of (24) and (21), it is seen that
c1 4n 11
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Note that a ,c0 ,c1 represents diophantine 3-tuple with property D (n 3)
Taking a ,c1 and employing the above procedure, it is seen that the triple a ,c1 ,c 2 where
c 2 9n 26
exhibits diophantine 3-tuple with property D (n 3)
Taking a ,c2 and employing the above procedure, it is seen that the triple a ,c2 ,c3 where
c3 16n 47
exhibits diophantine 3-tuple with property D (n 3)
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 1 , s 1,2,3,...
A few numerical examples are presented in Table 6 below:
Table 6: 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 5 (5, 4, 19) (5, 19, 44) (5, 44, 79) (5, 79, 124)
3 6 (6, 5, 23) (6, 23, 53) (6, 53, 95) (6, 95,149)
4 7 (7, 6, 27) (7, 27, 62) (7, 62, 111) (7, 111, 174)
5 8 (8, 7, 31) (8, 31, 71) (8, 71, 127) (8, 127,199)
Sequence 7:
4 PTn 2 Let a 3 n 3, c0 s 4n 2 Pn
It is observed that
2 2 2 2 ac0 s 4k 2n k 6k 3 s 2n k 3
Therefore, the pair a ,c0 represents diophantine 2-tuple with the property s2 4k 2n k 2 6k 3 .
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Let c1 be any non-zero polynomial such that
2 2 2 ac1 s 4k 2n k 6k 3 p (25)
2 2 2 c0c1 s 4k 2n k 6k 3 q (26)
Eliminating c1 between (25) and (26), we have
2 2 2 2 c0p aq c0 as 4k 2n k 6k 3 (27)
Introducing the linear transformations
p X aT , q X c0T (28)
in (27) and simplifying we get
2 2 2 2 X ac0T s 4k 2n k 6k 3
which is satisfied by T 1 , X s2 2n k 32
In view of (28) and (25), it is seen that
2 2 c1 n4s 4s 1 2s 2sk 3 3
Note that a ,c0 ,c1 represents diophantine 3-tuple with property Ds2 4k 2n k 2 6k 3
Taking a ,c1 and employing the above procedure, it is seen that the triple a,c1, c2 where
2 2 c2 n4s 8s 4 2s 4sk 312
2 2 exhibits diophantine 3-tuple with property Ds 4k 2n k 6k 3
Taking a ,c2 and employing the above procedure, it is seen that the triple a ,c2 ,c3 where
2 2 c3 n2s 3 2s 6sk 3 27
2 2 exhibits diophantine 3-tuple with property Ds 4k 2n k 6k 3
The repetition of the above process leads to the generation of sequence of diophantine 3-
tuples whose general form is given by a ,c1 ,c where
2 2 2 c1 n2s 1 2s 2 1sk 3 3 6 3 , 1, 2,3,......
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A few numerical examples are presented in Table 7 below:
Table 7: Numerical Examples
n s k D a,c ,c a,c0 ,c1 1 2 a ,c2 ,c3 a,c3 ,c4
2 2 2 124 (5, 40,81) (5, 81,132) (5,132,193) (5,193,264)
3 3 3 540 (6,126,20) (6,204,29) (6,294,396) (6,396,510)
4 4 4 1584 (7,288,41) (7,415,55) (7,556,711) (7,711,880)
5 5 5 3700 (8,550,738) (8,738,942) (8,942,1162) (8,1162,1398)
4. CONCLUSION
One may attempt for constructing sequences of Diophantine m -tuples involving other choices of higher degree number patterns.
REFERENCES
[1] I.G. Bashmakova, ed., Diophantus of Alexandria, “Arithmetics and the Book of Polygonal Numbers”, Nauka, Moscow, (1974).
[2] A.F. Beardon and M.N. Deshpande, “ Diophantine triples”, Math. Gazette 86, (2002), Pp:258-260.
[3] V. Pandichelvi, “Construction of the Diophantine triple involving polygonal numbers”, Impact J.Sci. Tech. 5(1), (2011), Pp: 7-11.
[4] M.A. Gopalan, G. Srividhya, “Two special Diophantine Triples”, Diophantus J.Math, 1(1), (2012), Pp: 23-27.
[5] M.A. Gopalan, V. Sangeetha and Manju Somanath, “Construction of the Diophantine triple involving polygonal numbers”, Sch. J. Eng. Tech. 2(1), (2014), Pp: 19-22.
[6] M.A. Gopalan, S. Vidhyalakshmi and S. Mallika, “Special family of Diophantine Triples”, Sch. J. Eng. Tech. 2(2A), (2014), Pp: 197-199.
[7] M.A Gopalan, K. Geetha, Manju Somanath, “On Special Diophantine Triples”, Archimedes Journal of Mathematics, 4(1), (2014), Pp: 37-43.
[8] M.A. Gopalan and V. Geetha, “Sequences of Diophantine triples”, JP Journal of Mathematical Sciences, Volume 14, Issues 1 & 2, (2015), Pp: 27-39.
[9] M.A. Gopalan and V. Geetha, “Formation of Diophantine Triples for Polygonal
Numbers t16, n to t25, n and Centered Polygonal Numbers ct16, n to ct25, n ”, IJITR, volume 3, Issue 1, ( December-January 2015), Pp:1837-1841.
[10] G. Janaki and S. Vidhya, “Construction of the diophantine triple involving Stella octangula number”, Journal of Mathematics and Informatics, vol.10, Special issue, (December 2017), Pp:89-93.
[11] G. Janaki and S. Vidhya, “Construction of the Diophantine Triple involving Pronic Number”, IJRASET, Volume 6, Issue I, (January 2018), Pp: 2201-2204.
Volume 8, Issue 3, 2020 http://aegaeum.com/ Page No: 1069 AEGAEUM JOURNAL ISSN NO: 0776-3808
[12] G. Janaki and C. Saranya, “Construction of the Diophantine Triple involving Pentatope Number”, IJRASET, Volume 6, Issue III, (March 2018), Pp: 2317-2319.
[13] J.Shanthi, M.A.Gopalan and Sharadha Kumar, “On Sequences of Diophantine 3- Tuples through Euler Polynomials”, IJAST, Vol 27, No.1, (2019), Pp:318-325.
[14] A.Vijayasankar, Sharadha Kumar, M.A.Gopalan, On Sequences of diophantine 3- tuples generated through Pronic Numbers, IOSR-JM, 15(5) ser.II, (Sep-Oct 2019), Pp:41-46.
[15] M.A.Gopalan , Sharadha Kumar, On Sequences of Diophantine 3-tuples generated through Euler and Bernoulli Polynomials, Tamap Journal of Mathematics and Statistics, Volume 2019,(2019), Pp:1-5.
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