International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 11 Issue: 10 | Oct 2024
p-ISSN: 2395-0072
www.irjet.net
Study on Non-Negative Integer Solutions of Exponential Diophantine Equation 512x + 1728y = z3 and 271x + 9y = z3 Parakram Singh Department of Mathematics, Planetskool Research centre, Sonipat, (HR) India ---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract – The aim of the present paper is to demonstrate
if resolvable, possible number of solutions and finally to find the complete results. Diophantine equations are frequently used in the field of Abstract algebra, Coordinate geometry, Group theory, Linear algebra, Trigonometry, Cryptography and asunder as well as we can define the number of rational points on a circle. In investigations on Diophantine equations of steps, two significant successes were scored only in the 20th century. It was proved by A. Thue [4]. Diophantus from Alexandria such equations are vociferated Diophantine equations. Mordell [12] studied the Diophantine equations. Acu [6] has studied the elementary solutions for the Diophantine equation of ax + by = cz. Suvarnamani et al. [3] analyzed two Diophantine equations 4x + 7y = z2 and 4x + 11y = z2. Sroysang [5] obtained the solutions for Diophantine Equations 2x + 3y = z2. Cohen [10] studied the Number Theory and also gave its tools he also dealt with many aspects of number theory, mainly the central theme being the solution of Diophantine equations. Cipu et al. [13] have revealed the number of extensions for a fixed Diophantine triple. Burshtein [14], considered the general equation of three consecutive prime integers of the form px + (p + 1)y + (p + 2)z = M3, where M represents a positive integer and p represents prime with p ≥ 2, x, y ≥ 1, & z ≤ 2, also he determined all solutions of the above exponential Diophantine equations. Janaki and Shankari [8] have discussed various implementable ways to tackle multivariable and multi-degree Diophantine problems and obtain the solution of these exponential Diophantine equations.
the problem of existence of the solution of exponential nonlinear Diophantine equation as there are no general methods to find solution with natural number. This is an attempt to find numerical solutions (if any) of the equations 512x + 1728 y = z3, and the 271x + 9y = z3, where (x, y, z) are non-negative integers.
Keywords: Exponential Diophantine equation, Number Theory, Non-Negative Integers Solution Mathematics Subject Classification (ASM): 11D61, 11D79
1. INTRODUCTION The theory of Number is an elegant branch of mathematics that primarily concerned with the study of non-negative integers, or counting numbers, and their properties as well as the solvability of equations in whole numbers. It has a veritably long and different history, and some of the topmost mathematicians of all time, similar as Euclid, Euler, and Gauss, have made significant benefactions to it. Hardy and Wright, [9] discussed a great diversity of different topics of theoretical number theory and found a remarkable selection of arithmetic problems treated with consummate clarity and distinction. Burton [7] suggested the study of Elementary & classical number theory and to impart some of the historical background in which the subject evolved. Niven et al. [11] have discussed the introduction to the theory of numbers and expand the binomial theorem, calculation methods for numerical and a public key cryptography section. Contains an outstanding set of problems. Baker, [1] and [2] provided comprehensive initiation to all the major branches of number theory including elements of cryptography and primality testing, an account of number fields in the arithmetic of elliptic curves. The particular type of Exponential Diophantine equation is analyzed and generalized by the method of Catalan's conjecture, its primary Cyclotomic units, and proof was given by Mihailescu [15].
The paper is organized as follows. Section 2 presents the Preliminary work of the paper by using Lemma and theorems. Section 3 and 4, presents the working strategy to solve the main exponential Diophantine equations of this paper. The conclusions about the obtained solutions are contain in section 5. The rest of the paper listed the related work as references.
2. Preliminary 2.1 Lemma: (Miheailescu’s Theorem) [15] The Dio-phantine equation ax − by = 1 has the unique solution (a, b, x, y) = (3, 2, 2, 3), where a, b, x and y are integers with min {a, b, x, y} > 1.
Algebraic equations with non-negative integer amounts having integer solutions are Diophantine equations. For finding the solution to these equations, there's no universal manner available yet, so the investigators are keenly interested in developing new techniques for unravelling these equations. While handling any cognate equation, three issues arise, that's whether the problem is resolvable or not;
© 2024, IRJET
|
Impact Factor value: 8.315
2.2 Lemma: There are no solutions in integer x, y, z with x, y, z > 0, of xn + yn = zn when n ≥ 3, also known as Fermat's Last Theorem, was proved by Wiles [16].
|
ISO 9001:2008 Certified Journal
|
Page 532