Intrinsic solutions of Diophantine Equation Involving Centered Square Number

https://doi.org/10.22214/ijraset.2023.49471

11 III

2023

March

ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538

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Intrinsic solutions of Diophantine Equation Involving Centered Square Number

1Assistant Professor, 2PG Student, PG and Research Department of Mathematics, Cauvery College for Women (Autonomous), Affiliated to Bharathidasan University, Trichy

18.

Abstract: The bi-quadratic Diophantine equation with five unknown parameters

researched for its quasi complex arithmetic values. A few correlations between the solutions and plethora of other figures notably the triangular, pronic, stella octagula and gnomonic notation are effectively portrayed.

Keywords: Bi-quadratic, Non-homogeneous, Integer solution, Diophantine equation, Centered square number.

I. INTRODUCTION

In introductory number theory, a centered square value is a way to portray the guesstimated number of dots together in square which would have perhaps one dot in the centre and every additional dot facing something in preliminary square strands. The range of centers in each centered square multitude is equal to the number of markings on a conventional square pattern with in a particular demographic block altitude of the centre dot. Centered square numbers, like figurate numbers in terms of appearance, have few if any applications in the real world, however they are sporadically studied in entertainment mathematics for their spectacular architectural and mathematically wonderful aspects. While isolated equations have indeed been explored throughout history as a kind of dilemma, the modernization of rigorous conceptions of Diophantine equations is a massive achievement of the twentieth century.[1-3] gives a detailed and self-explanatory study of Diophantine equations. For different techniques towards solving various Diophantine as well as exponential Diophantine equations. [4-19] have been referred. The purpose of research is to delineate nontrivial integral solutions to the five unknowns in the bi-qudratic Diophantine equation facilitated by

. Numerous incredibly interesting correlations between specific solutions and the numbers notably the triangular number, conjointly gnomonic number, pronic number, stella octangula number are proposed.

II. NOTATIONS

1) (2  n Gno n = Gnomonic number of rank n.

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2 2 2 4 4 ) )( 1) ( ( R l k n n H E   P. Saranya1 , K. Poorani2

–

2 2 2 4 4 ) )( 1) ( ( R l k n n H E   is

2 2 2 4 4 ) )( 1) ( ( R l k n n H E  

1)

2)

3)



4)

5)

6)

,  

T n m

7)

8)

9)

1) (2 2  n n So n =Stella Octangula number of rank n.

2 1) ( 1

 n mn C n m =Centered m-goal number of rank n.

2 2 1) (   n n CS n =Centered square number of rank n.

1) ( Pr   n n n =Pronic number of rank n.

) 2 2) 1)( ( (1

m n n

=Triangular number of rank n.

1 2  n n n W =Woodall number.

2 1) (2 2  n =Kynea number.

1 2 1 2  n Mn M =Double Mersenne number

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III. TECHNIQUE FOR ANALYSIS

The five unknown bi-quadratic equation is predicted to be,

The linear transformations,

Now that (1) has been simplified to,

As we compute (3), we observe an indefinite range of choices for various patterns. A.

Let

Utilizing (4) in

and using the method of factorization define,

Equalizing the real and the imaginary, we get

Hence the non-zero integral solutions to (1) are,

is a Woodall number.

Journal

International

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2 2 2 4 4 ) )( 1) ( ( R l k n n H E   (1)

             ) 4(1 ) 4(1 cd l cd k d c H d c E (2)

2 2 2 4 4 ) 4 4 4 )(4 1) ( ( ) ( ) ( R cd cd n n d c d c      2 2 2 2 2 ) )(8 1) ( ( ) ( 8 R cd n n d c cd    2 2 2 2 2 ) 1) ( ( R n n d c    (3)

Pattern: I

2 2 t s R   (4)

2) 1))( ( ( it s n i n id c     ) 2 1))( ( ( 2 2 t st i s n i n    2 2 2 2 1) ( 1) 2( 1) ( 2 t n i st n s n i nt nst i ns id c    

(3)

2 2 1) 2( nt st n ns c  2 2 1) ( 2 1)

nst s n d  

( t n

2 2 1) (2 2 1) (2 t n st s n E   2 2 1) 2(2 t st n s H  t s n n s n n k 3 2 2 4 ) 1) ( 8( 1) ( 4 4            4 3 2 2 2 2 1 4 1 8 1 24 t n n st n n t s n n  2 2 3 2 2 4 1) ( 24 ) 1) ( 8( 1) ( 4 4 t s n n t s n n s n n l   4 3 2 2 1) ( 4 ) 1) ( 8( t n n st n n  2 2 t s R   Observations 1) 2 4 ,1) ,

s s s H s s E 

2) R(2,2)-1=7

3)     23 1 1,1 1,1,1 5    R k

( ,1) , (

is a square integer.

is a double mersenne number.

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Some numerical examples are listed below.

and proceeding as in pattern 1 we get,

Realizing that the goal is to determine the integer result, we get from taking s=25S, t=25T

as we compute for the quasi integral solutions of (1) we get,

512 ©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | 4)    mod2 0 2 4 ,1 1,   s s Gno So s k 5)      mod1 0 2761 ,1 , ,1 ,   SGno S s s H s s E

TABLE 1 s t n E H k l R L.H.S= R.H.S 1 1 1 2 -2 4 4 2 0 2 1 1 7 -1 52 -44 5 2400 3 -2 1 -7 17 -236 244 13 -81120 -4 6 1 -68 28 3844 -3836 52 20766720 B. Pattern: II Rewriting (3) as, 2 2 2 2 2 ) 1) ( 1*( R n n d c    (5) Write 1 as, 2 2 2 25 20 15 1   (6) And take, ‘t’ as in (4)

(6)

(5)

20) (15 ) 1))( ( ( 2 i it s n i n id c         st n s n i nt nst i ns id c 1 30 1 15 15 30 15 2 2 2         2 2 2 2 1 20 20 40 20 1 15 s n nt i nst ns i t n i      2 1 20 1 40 t n st n i  Equalizing the real and imaginary part, we get st nst t nt s ns d st nst t nt s ns c 40 10 15 35 15 35 30 70 20 5 20 5 2 2 2 2 2 2 2 2       

ST nST T nT S nS d ST nST T nT S nS c 1000 250 375 875 375 875 750 17500 500 125 500 125 2 2 2 2 2 2 2 2        Consequently,

ST nST T nT S nS E 1750 2000 125 750 125 750 2 2 2 2    ST nST T T S nS H 250 1500 875 1000 875 1000 2 2 2 2    3 2 3 2 2 2 2 4 2 1500000 1500000 656250 109375 4(1 ST n T S n T S n S n k    3 2 2 4 4 2 1062500 2906250 484375 109375 nST T nS nS T n  2 2 4 4 3 112500 187500 484375 1062500 T S S nT T nS    ) 187500 218750 218750 4 3 3 T ST T S  3 2 3 2 2 2 2 4 2 1500000 1500000 656250 109375 4(1 ST n T S n T S n S n l    3 2 2 4 4 2 1062500 2906250 484375 109375 nST T nS nS T n   

Using

in

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Some numerical examples are listed below.

Utilizing

and proceeding as in pattern-II we get

513 ©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | 2 2 4 4 3 1125000 187500 484375 1062500 T S S nT T nS  ) 187500 218750 218750 4 3 3 T ST T S   2 2 625 625 T S R   Observations 1) 3 8 ,1,1) ( ,1,1) ( s s l s k   is a cubical integer. 2)   0 625 250 125 ,1,1 16,   s s Gno T s E 3)       s CS s s sH s s E s s k 22500000 ,1 , 500 ,1 , 3000 ,1,1 2    mod1187504 0 22062500   s Gno 4)     895 145 1,1 ] 1,1,1 (1,1,1) [    R H E is a Woodall number. 5)   1087 663 ] 1,1,1 [  H is a Kynea number. 6)    mod875 0 250 125 ,1 12,  s s Gno T s R

TABLE 2 s t n E H k L R L.H.S=R.H.S 1 1 1 -250 -1750 -2999996 3000004 1250 -9375000000000.00 43 3 67625 42125 27986250042798624996 15625 17764709472656300000 6 4 2215001005009637999996 9638000004 32500 -1018013750000000000002 3 5 30125 671253598249996 3598250004 812519478339257812500000.00 C. Pattern: III ‘1’ in (6) can

as, 2 2 2 17 15 8 1   (7)

2) 15)( 1))(8 ( 1*( iy x i n i n id c      Equating the real and imaginary part, we get st nst t nt s ns c 16 46 15 7 15 7 2 2 2 2     st nst t nt s ns d 30 14 8 23 8 23 2 2 2 2    Realizing that the goal is to

the integer result, we get from taking s=17S, t=17T ST nST T nT S nS c 275 782 225 119 225 119 2 2 2 2     ST nST T nT S nS d 510 238 136 391 136 391 2 2 2 2    Consequently, as we compute for the quasi integral of (1), we get ST nST T nT S nS E 782 1020 89 272 89 272 2 2 2 2    ST nST T nT S nS H 233 544 361 510 361 510 2 2 2 2   

also be written

(7) in (5)

determine

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Some numerical examples are listed below.

Utilizing (8) in (5) and proceeding as in pattern-II, we get

that the goal is to determine the integer result, we get from taking s=10S, t=10T

514 ©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | 3 2 3 2 2 2 2 4 2 277440 277440 279174 46529 4(1 ST n T S n T S n S n k    T nS T nS nS T n 3 2 2 4 4 2 98912 671874 104159 46529   2 2 4 4 3 199920 30600 104159 98912 T S S nT nST   4 3 3 30600 77758 77758 T ST T S  ) 3 2 3 2 2 2 2 4 2 277440 277440 279174 46529 4(1 ST n T S n T S n S n l    T nS T nS nS T n 3 2 2 4 4 2 98912 671874 104159 46529   2 2 4 4 3 199920 30600 104159 98912 T S S nT nST   ) 30600 77758 77758 4 3 3 T ST T S   2 2 289 289 T S R   Observation 1)     959 61 ,1) , ( ,1 , 2   S S S H S S E is a Kynea number. 2) 2R(1,1) is a perfect square. 3) 0(mod510) 510 1020 ,1) , ( ,1) , ( 2,     S S Gno C S S H S S E 4) 0(mod119) 119 238Pr ,1) , (   S S Gno S S E 5)     0 361 2127 325 ,1 ,1,1 6,    s T s R s E s

TABLE 3 s t n E H K L R L.H.S=R.H.S 1 1 1 -238 -782 -554876 554884 578 -370753059840.00 2 2 1 -952 -3128 -8878076 8878084 2312 -94912783319040.00 2 2 2 -5032 -5304 -2811388 2811396 2312150278573588480.00 3 3 1 -2142 -7038 -44945276 44945284 5202 -2432510825610240

Pattern: IV

1 as, 2 2 2 10 8 6 1   (8)

D.

Write

st nst t nt s ns c 12 28 8 2 8 2 2 2 2 2     st nst t nt s ns d 16 4 6 14 6 14 2 2 2 2   

ST nST T nT S nS E 280 320 20 120 20 120 2 2 2 2    ST nST T nT S nS H 40 240 140 160 140 160 2 2 2 2    3 2 3 2 2 2 2 4 2 38400 38400 16800 2800 4(1 ST n T S n T S n S n k    T nS T nS nS T n 3 2 2 4 4 2 27200 74400 12400 2800  

Realizing

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Some numerical examples are listed below.

IV. CONCLUSION

We have constructed an infinite number of non intriguing numerical solutions to the bi-quadratic Diophantine equation with five unknowns. For the equation beneath cognizance, various pattern of solutions can always be obtained.

REFERENCES

[1] L.E.Dickson, History of the Theory of Numbers, Chelsea Publishing Company, New York, Vol II, 1952.

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[3] L.J. Mordell, Diophantine equations, Academic Press, New York, 1969.

[4] P Saranya , G. Janaki , Ascertainment on the integral solutions of the Bi-quadratic Diophantine Equation

Parishodh Journal, volume 9,Issue III, ISSN NO: 2347-6648, March 2020.

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2, Issue 3, PP: 396-397, 2016.

[6] G Janaki , P Saranya, On the Ternary cubic Diophantine Equation

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[7] P Saranya, G Janaki, On the Quintic Non-Homogeneous Diophantine Equation

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515 ©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | T S T S S nT nST 3 2 2 4 4 3 5600 28800 4800 12400 27200    ) 4800 5600 4 3 T ST 3 2 3 2 2 2 2 4 2 38400 38400 16800 2800 4(1 ST n T S n T S n S n l    3 3 2 2 4 4 2 27200 27200 74400 12400 2800 nST T nS T nS nS T n    ) 4800 5600 5600 28800 4800 12400 4 3 3 2 2 4 4 T ST T S T S S nT    2 2 100 100 T S R   Observation 1) 287 7 ,1)] , ( [ 2 2   S S S S H is a kynea number. 2) 1000 (1,1,1)] (1,1,1) (2,2) 2[    E H R is a cube root. 3) SSo R S E S S S l 11200 (1,1) 576 ,1,1) ( 480 ,1) , ( 2 0(mod23596) 4400  SGno 4) 16 (1,1,1)] (1,1,1) 2[   l k is a perfect square. 5)   0 230 90 140Pr ,1,1    s s Gno s E 6)    mod100 0 75 25 ,1 10,  s T s R s

TABLE 4 s t n E H K l R L.H.S=R.H.S 1 1 1 -40 -280 -76796 76804 200 -6144000000 2 1 2 60 -1580 -2492796 2492804 500 -6232000000000.00 6 2 1 4000 -4000 4 4 4000 0 -3 4 2 2500 7500 -49999996.00 50000004 2500 -3125000000000000.00

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  3 2 2 4 4 40 p w z y x  , International Journal of Engineering

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ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.538

Volume 11 Issue III Mar 2023- Available at www.ijraset.com

[8] P Saranya, G Janaki, On the Exponential Diophantine Equation,

Issue 11, PP:1042-1044, 2017.

[9] P Saranya, G Janaki, Explication of the Transcendental Equation,

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[10] P Saranya, G Janaki, Ascertainment On The Exponential Equation

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[17] C. Saranya, G. Janaki “ Integral Solutions Of The Homogeneous Quintic Diophantine Equatio x5 -y 5 -x 2 y 2(x-y)=972(x-y)(z+w)2 p 2, International Journal for Scientific Research & Development, Volume 5, Issue 7, September 2017. Pg. No. 75-77, Impact Factor: 4.396, ISSN No: 2321-0613.

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[19] S.Vidhya, G.Janaki “Observations on Ternary Quadratic Equation z2 = 82x2 + y2”, International Research Journal of Engineering and Technology, Vol 4, Issue 3, March 2017, Pg.no: 1239-1245, Impact Factor: 5.181, ISSN NO: 2395-0056.

2 3 36 z y x   ,

Research

of

and Technology, Vol 4,

International

Journal

Engineering

  3 2 2 2 2 3 1 t m s r pq q p p       , Adalya Journal,

c b b c c p p p a p   ) ( 2 3 , Compliance Engineering Journal, Vol 10, Issue 8,

2 2 2 972 7 5 z y x   ”, Jamal Academic Research Journal – An

  2 2 2 72 9 3 3 11 6 z y x xy y x      ”,

2, February 2016. Pg. No.