Page 154 - ASBIRES-2017_Preceedings

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Proceedings of the 9 Symposium on Applied Science, Business & Industrial Research – 2017
ISSN 2279-1558, ISBN 978-955-7442-09-9

Solution to Special Diophantine Equations Using the Remainder Theorem

Nimasha BGBS, Piyadasa RAD
Department of Mathematical Sciences, Wayamba University of Sri Lanka
nimashabgbs91@gmail.com

ABSTRACT

In this paper, we solve special Diophantine equations which are related to a proof of
Fermat’s last theorem using the Remainder Theorem. Firstly, we discuss about what is
meant by linear Diophantine Equations. In our research, we consider the solutions of the
Diophantine Equations of higher orders which are related to a proof Fermat’s Last
Theorem (FLT) and have been used to prove the FLT together with the Remainder
Theorem. Solutions of the Diophantine equations of higher orders and using the
Remainder theorem for finding the solutions of those Diophantine equations is
questionable as far as we understand. As the first task of this research, Remainder theorem
is proved so that in finding usual remainder there is no need of the substitution of number
to the variable. In the next step, Diophantine equations which are related to Fermat’s last
theorem are solved by using the new proof of the Remainder Theorem. In this project
work, we prove that the correct use of the Remainder Theorem can be applied even when
the numbers are co-prime. In conclusion, we understand that solving the Diophantine
Equations their parametric equations are very useful.

KEYWORDS: Diophantine equations, Polynomial equations, Remainder theorem

1 INTRODUCTION are solved. These Diophantine equations are

directly related to Fermat equation = +
1.1 Diophantine Equation
for > 2. When = 2 there are infinitely

Diophantine equation is a polynomial many solutions and it was famous as
equation with two or more unknowns and it Pythagorean triples (, , ). For larger
is considered for only the integer solution. integer values of > 2, Fermat’s Last
A linear Diophantine equation means an Theorem (Initially claimed in 1637 by
equation between two sums of monomials of Fermat and proved by Wiles in 1995) sates
degree zero or one. (Monomial is a there are no positive integer solutions
polynomial which has only one term). (, , ).The simplest Linear Diophantine
equation takes the form + = , where
The word Diophantine refers to the
Hellenistic mathematician of the 3rd , and are integers. This Diophantine
century, Diophantus of Alexandria, who equation has a solution (where x and y are
made a study of such equation and was one integers) if and only if c is a multiple of
of the first mathematician to introduce the greatest common divis
symbolism into algebra. or of a and b. Moreover, if (x, y) is a
solution, then the other solutions have the
The mathematical study of Diophantine form ( + , − ), where k is an arbitrary
problem that Diaphanous initiated is now integer, and u and v are the quotients of a
called Diophantine analysis. Diophantine and b (respectively) by the greatest common
equations are usually difficult to solve. The divisor of a and b.
most difficult Diophantine equation is given
by Fermat’s Last Theorem in which we have 1.2 The Equations Related to the Farmat’s
to show that the equation = + is not Last Theorem

satisfied by positive co-prime integers In the following, we consider the
, , integers for any > 2. In this research Diophantine equations related to Farmat’s
work, two kinds of Diophantine equations

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