AEGAEUM JOURNAL ISSN NO: 0776-3808

CONSTRUCTION OF SEQUENCES OF DIOPHANTINE 3-TUPLES THROUGH THE RATIO OF PENTATOPE AND TETRAHEDRAL

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 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 a1 ,a 2 ,a 3 ,.....,a m is said to

have the property Dn,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 Dn. Many Mathematicians considered the construction of different formulations of diophantine triples with the property Dn 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

nn 1n  2n  3 1. PT   Pentatope of rank n n 24 nn 1n  2 2. P3   of rank n n 6

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3. METHOD OF ANALYSIS

Sequence 1:

PTnk3 PTnk3 Let a  4 3  n  k , c0  4 3  n  k Pnk3 Pnk3

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  ak (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 ,cs1 ,cs where

2 2 cs1  s n  s  2sk , 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:

PTn1 PTn5 Let a  4 3  n  2, c0  4 3  n  2 Pn1 Pn5

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  a4 (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 ,cs1 ,cs where

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2 2 cs1  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 PTn6 Let a  4 3  n  3, c0  4 3  n  3 Pn Pn6

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  a9 (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 ,cs1 ,cs where

2 2 cs1  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 n2 a  4 3  n  3, c0  4 3  n 1 Pn Pn2

It is observed that

2 2 ac0  2k  2n  k  2k  2  n  k 1

Therefore, the pair a ,c0  represents diophantine 2-tuple with the property D2k  2n  k2  2k  2 .

Let c1 be any non-zero polynomial such that

2 2 ac1  2k  2n  k  2k  2  p (13)

2 2 c0c1  2k  2n  k  2k  2  q (14)

Eliminating c1 between (13) and (14), we have

2 2 2 c0p  aq  c0  a2k  2n  k  2k  2 (15)

Introducing the linear transformations

p  X  aT , q  X  c0T (16)

in (15) and simplifying we get

X2  abT2  2k  2n  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  2n  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  2n  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  2n  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 ,cs1 ,cs where

2 2 cs1  s n  2s 1k  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 n1 n2 a  4 3  n  2, c0  4 3  n 1 Pn1 Pn2

It is observed that

2 2 ac0  2k  3n  k  2  n  k

Therefore, the pair a,c0  represents diophantine 2-tuple with the property

D2k  3n  k2  2 .

Let c1 be any non-zero polynomial such that

2 2 ac1  2k  3n  k  2  p (17)

2 2 c0c1  2k  3n  k  2  q (18)

Eliminating c1 between (17) and (18), we have

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2 2 2 c0p  aq  c0  a2k  3n  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  3n  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  3n  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  3n  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  3n  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  3n  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 ,cs1 ,cs where

2 2 cs1  s n  2s 1k  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 n1 a  4 3  n  3, c0  4 3  n  2 Pn Pn1

It is observed that

2 ac0  n  3  n  3

Therefore, the pair a ,c0  represents diophantine 2-tuple with the property Dn  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  an  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 ,cs1 ,cs where

2 2 cs1  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  2n  k  6k  3 s 2n  k  3

Therefore, the pair a ,c0  represents diophantine 2-tuple with the property s2 4k  2n  k 2  6k  3 .

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Let c1 be any non-zero polynomial such that

2 2 2 ac1  s 4k  2n  k  6k  3 p (25)

2 2 2 c0c1  s 4k  2n  k  6k  3 q (26)

Eliminating c1 between (25) and (26), we have

2 2 2 2 c0p  aq  c0  as 4k  2n  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  2n  k  6k  3

which is satisfied by T 1 , X  s2 2n  k  32

In view of (28) and (25), it is seen that

2 2 c1  n4s  4s 1 2s  2sk  3 3

Note that a ,c0 ,c1  represents diophantine 3-tuple with property Ds2 4k  2n  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  n4s  8s  4 2s  4sk  312

2 2 exhibits diophantine 3-tuple with property Ds 4k  2n  k  6k  3

Taking a ,c2  and employing the above procedure, it is seen that the triple a ,c2 ,c3  where

2 2 c3  n2s  3  2s  6sk  3 27

2 2 exhibits diophantine 3-tuple with property Ds 4k  2n  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 ,c1 ,c  where

2 2 2 c1  n2s   1  2s  2 1sk  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 ”, IJRASET, Volume 6, Issue I, (January 2018), Pp: 2201-2204.

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[12] G. Janaki and C. Saranya, “Construction of the Diophantine Triple involving ”, 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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