On the Sextic Diophantine Equation with Three Unknowns

International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-06, Sep 2018 On The Sextic Diophantine Equation With Three Unknowns

1R. Anbuselvi, 2K.Kannaki 1Associate Professor of Mathematics, ADM College for women (Autonomous), Nagapattinam, Tamilnadu, India 2Lecturer of Mathematics, Valivalam Desikar Polytechnic College, Nagapattinam, Tamilnadu, India ABSTRACT - Sextic Diophantine equation with three unknowns has been analyzed for its non- zero integral solutions. Two different patterns of solutions and their corresponding properties are obtained. Keywords: Sextic equation with three unknowns, Integral solutions. Notations:

1. , Octahedral number.

2. , Stella Octangula number.

3. , Pronic number of rank n.

4. Gnomonic number of rank p.

5. Hauy Rhombic dodecahedral number.

6. , four dimensional figurate number of rank n.

7. , Nexus number of rank n.

8. , Centered Hexagonal number of rank n.

9. , Kynea number of rank n.

10. , Heptadecagonal number of rank n.

I. INTRODUCTION [15-18], Sextic equations with three unknowns are studies for their non-zero integer solutions. Diophantine equations, homogeneous and non- In this statement, however another fascinating Sextic homogeneous have aroused the significance of frequent Diophantine equation with three unknowns mathematicians since ancient times as can be seen from [1, is well thought-out for integral solutions. Two 2]. The problem of result all integral solutions of a different patterns of solutions and their related properties Diophantine equation with three or more variables and are obtained. degree as a minimum three in universal, in presents a superior deal of difficulties. There is a massive common II. 2. METHOD OF ANALYSIS: theory of homogeneous quadratic equations with three The equation below concern is variables [1-5]. Cubic equations in two variables reduce into the theory of elliptic curves which is especially --- (1) industrial theory but unmoving an vital matter of modern Passing on the transformations research [6, 7]. A bundle is known in relation to equations in two variables in superior degrees. For equations with --- (2) more than three variables and degree at least three, in (1) leads to incredibly tiny is known. It is meaning to communication that undesirability appears in equations, smooth possibly at --- (3) degree four and degree five with comparatively little coefficients. In this circumstance one may refer [9-14]. In

577 | IJREAMV04I0642117 DOI : 10.18231/2454-9150.2018.0779 © 2018, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-06, Sep 2018

The equation (3) is solved throughout special approaches and they, one obtains separate sets of solutions is (1) Case 1: Imagine that Substituting the above values of u and v in --- (4) equation (2), and hence the non-zero integral solutions of √ √ (1) are Mark

Where n = 1, 2, 3

--- (5) use (5) & (4) in (3) and applying the way of factorization, classify --- (7) √ ( √ ) √

--- (6)

Equating the real and imaginary parts, we have Properties:

2) ( )

4) ( )

5) ( )

6) ( ) ( )

8) 9) and is a perfect squares. 10) is a perfect biquadratic integers. Case 2: Equ (3) can be written as two different ways --- (8) and --- (9) Another time mark 12 as

√ √ --- (10) and as well 1 as

√ √ --- (11) use (4), (11), (10) in (8) and applying the method of factorization, classify

√ { √ ( √ )( √ ) } --- (12)

Mark 4 as

√ √ --- (13) and as well as 3 as

√ √ --- (14) use (4), (15) and (14) in (9) and applying the method of factorization, classify

√ {( √ ) √ ( √ ) } --- (15)

Equating the real and imaginary part of the above equations (12) and (15), we have the similar outcome

Substituting the above values of u and v in equation (2), and hence the non-zero integral solutions of (1) are

578 | IJREAMV04I0642117 DOI : 10.18231/2454-9150.2018.0779 © 2018, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-06, Sep 2018

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III. CONCLUSION [14] Anbuselvi R, Kannaki K, On ternary Quadratic Equation We have accessible two different patterns of solutions for IJSR Sep 2016: 5(9); 42-48. the Sextic Diophantine equation with three unknowns . To conclude one may search for other [15] Anbuselvi R, Kannaki K, On ternary Quadratic patterns of solutions and their corresponding properties. Diophantine Equation IERJ Feb 2017: 3(2); 52-57. REFERENCE [16] Anbuselvi R, Kannaki K, On the Heptic Diophantine [1] Dickson LE; History of Theory of Numbers, Volume 2, Equation with Three Unknowns Chelsea Publishing Company, New York, 1952. Paripex – 1JR May 2017 6(5); 622-624. [2] Mordell LJ; Diophantine Equations, Academic Press, [17] Anbuselvi R, Kannaki K, On the Homogeneous New York 1969. Biquadratic Equation with Four Unknowns

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