SOLUTION TO EQUAL SUM OF FIFTH POWER DIOPHANTINE EQUATIONS - A NEW APPROACH
By
Narinder Kumar Wadhawan
Civil Servant, Indian Administrative Service Retired, House No. 563, Sector 2, Panchkula, Haryana, India-134112,
Email: narinderkw@gmail.com
(Received: November 23, 2022; In format: November 17, 2022; Revised: February 09, 2023; Accepted; February 17, 2023)
DOI: https://doi.org/10.58250/jnanabha.2023.53116
Abstract
Purpose of writing this paper is to introduce simple parametric solutions to quintic Diophantine equations 5.n.m where integer n > 2 and integer m > 3. Methodology applied is writing numbers in algebraic form as aI x + bI with variable x, then writing fifth power Diophantine equation, in algebraic form with one variable and then transforming it to a linear equation by vanishing its four terms. For achieving this purpose, values to aI and bI of algebraic numbers are assigned so as to vanish constant term and coefficient of fifth power of x. Then equating with zero the coefficient of second power and coefficient of x, vanishes other two terms. These operations yield two relations between various aI and bI and also a linear equation in x. On putting the value of x obtained from this linear equation in given Diophantine equation, provides solution. Paper provides a single direct parametric solution to all quintic Diophantine equations 5.n.n where ∞ > n > 5 and is simple, easily comprehensible and didactic.
2020 Mathematical Sciences Classification: 11D4.
Keywords and Phrases: Integers, Rational quantity, Linear equation, Diophantine equation of fifth power.