On Integer Solutions of the Ternary Quadratic Equation

https://doi.org/10.22214/ijraset.2023.49684

11 III

2023

March

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

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

On Integer Solutions of the Ternary Quadratic Equation 2

332 2 2 3 2 3 n ar r a

G. Janaki1 , M. Aruna2

1Associate Professor, PG and Research Department of Mathematics, Cauvery College for Women (Autonomous), Affiliated to Bharathidasan University, Trichy – 18.

2PG Student, PG and Research Department of Mathematics, Cauvery College for Women (Autonomous), Affiliated to Bharathidasan University, Trichy – 18.

Abstract: Analysis is conducted on the non-trivial different integral solution to the quadratic equation 2 2 2 332 2 3 3 n ar r a 

. We derive distinct integral solutions in four different patterns. There are a few intriguing connectionsbetweenthesolutionsanduniquepolygonalnumbersthatarepresented.

Keywords:Quadraticequation,integralsolutions, polygonalnumbers,specialnumbers, squarenumber.

I. INTRODUCTION

Number theory is a vast and fascinating field of mathematics concerned with the properties of numbers in general and integers in particular as well as the wider classes of problems that arise from their study. Number theory has fascinated and inspired both amateurs and mathematicians for over two millennia. A sound and fundamental body of knowledge, it has been developed by the untiring pursuits of mathematicians all over the world. The study of Number theory is very important because all other branches depend upon this branch for their final results. The older term for number theory is “arithmetic”. During the seventeenth century. The term “Number theory” was coined by the French mathematician Pierre Fermat who is consider as the “Father of modern number theory”. The first scientific approach to the study of integers, that is the true origin of the theory of numbers, is generally attribution to the Greeks. Around 600BC Pythagoras and his disciples made rather thorough studies of this integer A Greek mathematician, Diaphantus of Alexandria was able to solve equations with two or three unknowns. These equations are called Diophantine equation the study of these is known as “Diophantine analysis”. The basic problem is representation of an integers n by the quadratic form with the integral values x and y .

A linear Diophantine equation is an equation between two sums of monomial of degree zero (or) one. In 628 AD, Brahmagupta an Indian mathematician gave the first explicit solution of the quadratic equation. The word quadratic is derived from the Latin word quadrates for square. The quadratic equation is a second-order polynomial equation in a single variable x. There are several different ternary quadratic equations. To comprehend something in more detail is [1-4] visible For the non-trivial integral answers to the ternary quadratic equation [5-7] has been researched. For numerous ternary quadratic equation [8-10] has been cited. In this article, we investigate another intriguing ternary quadratic equation 2 2 2 332 2 3 3 n ar r a   and obtain several non-trivial integral patterns. A few intriguing connections between the solutions and unique polygons, rhombic, centered and Gnomonic number are displayed

A. Notations

 n m T , Polygonal number of rank n with size m

n RD = Rhombic dodecagonal number of rank n

5

n P = Pentagonal Pyramidal number of rank n

n TO = Truncated octahedral number of rank n

4

n P = Square Pyramidal number of rank n

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  



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n CC = Centered cube number of rank n

n SO = Stella octangula number of rank n

 n Gno Gnomonic number of rank n

II. METHOD OF ANALYSIS

In order to find a non-zero distinct integral solution to the ternary quadratic equation

On substitution of the linear transformations,

The following list offers four patterns for solution (3). The appropriate values of a and r are derived after the vales of

are known by using (2).

the factorization method and substituting (4) and (5) in (3),

and

are determined by equating the real and imaginary parts

Substituting the above values of f and g in equation (2), we get

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2 2 2

a  

332 2 3 3 n ar r

(1)

g f a  

r  (2) in (1) leads to 2 2 2 83 2 n g f   (3)

, g f

and g

A. Pattern: I In (3), 2 2 2 83 2 n g f   Assume that 0 , , 2 2 2    s h s h n (4) Write 2) 2)(9 (9 83 i i   (5)

) 2 ( g i f  ) 2 ( g i f = 2 2 ) 2 ( ) 2 2)( 2)(9 (9 s i h s i h i i   (6)

of f

hs s h s h f f 4 18 9 ) , ( 2 2   (7) hs s h s h g g 18 2 ) , ( 2 2    (8)

f

Applying

The values

g

2 2 20 14 10 ) , ( s hs h s h a   2 2 16 22 8 ) , ( s hs h s h r  2 2 2 ) , ( s h s h n   Properties perfectsquare isa h h n h h r h h a ) , ( 27 ) , ( 5 ) , ( 1.8  . 0(mod19). 20 4 1) , ( 2. 12,     h h h Gno T RD h h ha 0(mod40) 8 ,1) ( 2 ,1) ( ,1) ( 3. 6,   h T h n h r h a 0(mod9) 11 3 2 ,1) ( ,1) ( 4. 8, 5   h h h Gno T P h a h hn nastynumber isa n(1,1) 5.18

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1550 ©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | B. Pattern: II In (3), 2 2 2 83 2 n g f   Take the linear transformations T A g T A n 83 2     (9) T A g T A n 83 2   (10) Substituting (9) or (10) in (3), we get ) 166 81( 2 2 2 T A f  (11) Write F f 9  (12) Substituting (12) in (11), we get 2 2 2 166T A F  2 2 2 166T F A   (13) This is the standard form of Pell equation 2 2 2 z Dy x   The solutions to (13) that correspond to this are 2 2 2 2 166 166 2 s h A s h F hs T     (14) Substituting (14) in (9) and (12), we get 2 2 2 2 2 2 166 166 9 1494 4 166 s hs h g s h f s hs h n        (15) Substituting (15) in (2), we get 2 2 2 2 2 2 4 166 ) , ( 10 166 1328 ) , ( 8 166 1660 ) , ( s hs h s h n s hs h s h r s hs h s h a       (16) Properties: 5 ) , ( ) , ( ) , ( 1 4, perfectsquare a is T h h n h h r h h a h   0(mod413) 8 913 83 ,1) ( ,1) ( 2. 8,    h h h Gno T TO h hr h n . 405 ) , ( ) , ( 3. 4, perfectcube a is hT h h hn h h hr h   . (1,1) 4. palindromenumber is a 0(mod402). 413 166 ,1) ( ,1) ( 5. 16,  h h Gno T h n h r

International Journal for Research in Applied Science & Engineering Technology (IJRASET)

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

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1551 ©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | C. Pattern: III Consider equation (3) as 2 2 2 2 2 2 81 g n n f  (17) Factorizing (17) we have, ) )( 2( ) 9 )( 9 ( g n g n n f n f    0 , 9 ) 2( 9      S S H n f g n g n n f (18) Expressing this as a system of simultaneous equation 0 ) 2 (9 2   H S n Hg Sf (19) 0 ) (9    S H n Sg Hf (20) Using the cross multiplication method, we get 2 2 2 ) , ( S H S H n n    (21) 2 2 2 2 18 2 ) , ( 9 4 18 ) , ( S HS H S H g g S HS H S H f f        (22) Substituting (22) in (2), we get 2 2 2 2 2 2 2 ) , ( 10 14 20 ) , ( 8 22 16 ) , ( S H S H n S HS H S H r S HS H S H a      (23) Properties perfectsquare a is T H H n H H r H H a H4, 12 ) , ( ) , ( ) , ( 1.   0(mod2) 20 4 ,1) ( ,1) ( 2. 20,   H H Gno T H r H a 0(mod8) 30 3 18 6 2 6 ,1) ( )) 1, ( ( 3. 4, 22, 7, 17, 19, 4      H H H H H H T T T T T P H a H H n H . (1,1) 4. number woodall is a . (1,1) (1,1)) 5.2( cube perfect isa r a D. Pattern: 4 In addition to (18), (3) can be expressed as 0 , 9 ) 2( 9      S S H n f g n g n n f (24) Expressing this as a system of simultaneous equation 0 ) 2 (9 2    M S n Hg Sf (25) 0 ) (9   S H n Sg Hf (26) Using the cross multiplication method, we get ) (2 ) , ( 2 2 S H S H n n    (27)

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III. CONCLUSION

has four different patterns of non- zero distinct integral solutions, which we described in this paper. For other quadratic equation , one can look for other patterns of non-zero integer unique solutions and their accompanying features.

The ternary quadratic equation

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[1] Dickson L.E., “History of the theory of numbers”, Chelsia Publishing Co., Vol II, New York, 1952

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[3] Mordell L .J, “Diophantine Equations”, Academic Press, London 1969

[4] Telang S. G., “Number Theory”, Tata Mc Graw-Hill Publishing Company, New Delhi , 1996

[5] Janaki G, Saranya C, “Integral Solutions of Binary Quadratic Diophantine Equation

Journal of Scientific Research in Mathematical and Statistical Sciences , Vol 7, Issue II, Pg.No152-155, April 2020.

[6] Janaki G, Saranya C, “Observations on Ternary Quadratic Diophantine Equation

Journal of Innovative Research and science, Engineering and Technology, Volume 5, Issue 2, February

[7] Janaki G, Saranya C, “ On the Ternary Quadratic Diophantine Equation

, Vol 2, Feb 2016.

[8] Janaki G and Vidyalakshmi S, “Integral solutions of xy+x+y+1 = z2 - w2” , Antartica J.math , 7(1), 31-37, (2010).

[9] M.A. Gopalan and S.Vidyalakshmi, “Quadratic Diophantine equation with four variables x2+y2+xy+y=u2+v2uv+u-v”.

2008

[10] Janaki G, Radha R, “On Ternary Quadratic Diophantine Equation

Science and Engineering Technology, Vol 6, Issue I, January 2018.

International

1552 ©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | 2 2 2 2 18 2 ) , ( 9 4 18 ) , ( S HS H S H g g S HS H S H f f      (28) Substituting (28) in (2), we get ) (2 ) , ( 10 14 20 ) , ( 8 22 16 ) , ( 2 2 2 2 2 S H S H n S HS H S H r S HS H S H a      (29) Properties perfectcube isa H H n H H r H H a ) , ( 2 ) , ( ) , ( 1.   perfectcube isa n(1,1) 72 2. isaducknumber n r (1,1) 12 (1,1) 3.4  0(mod16) 15 3 ,1)) ( ,1) ( ,1) ( ( 4. 28,   H H H Gno T CC H n H a H r H 0 4 ) 1, ( 5. 7,    H H T SO H H a

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