Page 144 - ASBIRES-2017_Preceedings

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

New Simple Proofs of Fermat’s Last Theorem for n=3

Wickramasinghe WAAD, Piyadasa RAD
Department of Mathematical Sciences, Wayamba University of Sri Lanka
ayesha91wickramasinghe@gmail.com

ABSTRACT

In this paper, we have proved Fermat’s Last Theorem (FLT) for = in two ways
using new elementary methods. In the first proof, we have transformed the corresponding

Fermat equation to the equation = + where = and = (> 0). In the second

step, the above rational number equation was transformed to a polynomial equation of
where = − . Since the usual method of solving this cubic failed, we have used a simple
inequality to derive a contradiction. In the second proof of the same theorem, we have
replaced by / and transformed to the cubic in to an equation involving integers
and . First, we have looked for the integral solution of and then proving Fermat’s last
theorem for = , we have discussed briefly, the possibility of applying the same method
that we applied for = , in particular to the Fermat equation of the index , that is, for
any odd prime. In the second step, we have looked for rational number solutions of the
form / where ≠ . Hence, we have proved FLT for = .

KEYWORDS: Elementary mathematics, Fermat’s last theorem, Method of infinite descent

1 INTRODUCTION respect, FLT for = 3 is proved using
elementary Mathematics. The method used
Fermat’ Last Theorem (FLT) was in these cases must be extended for any >
written by Pierre de Fermat in 1637. It 2.
became public in 1670 without a proof. FLT

states that + = has no non trivial The traditional proof of FLT for =

integer solutions for , and when > 2. 4 is based on the method of infinite descent
It is well known that FLT despite its rather of Fermat and it follows from Fermat’s
simple statement had been very difficult to proof of the theorem that area of a right
prove for a general exponent and in fact, a triangle cannot be a square of an integer
formal complete proof remained elusive (Edwards, 1977 & Ribenboim, 1991). FLT
until 1995 when Andrew Wiles put forward is well known as the most famous and the
one based on elliptic curves. However, most most difficult theorem in number theory and
of the mathematicians are of the view that several research studies have been
Wiles’ proof is very difficult and very conducted over the years by amateurs
lengthy. But according to famous Harvey (Piyadasa, 2011), mathematicians
Friedman grand conjecture, it is possible to (Ribenboim, 1991) and philosopher
find simple proofs for all exponents using (McLarty, 2010). As far as we know, there
elementary mathematics. Also, a is no any other method to prove FLT for
philosopher, McLarty (2010) of Ease = 4 except the method of infinite descent.
Western Reserve University claims that FLT Fermat’s theorems have been studied by
can be proved using elementary amateurs (Piyadasa, 2007 and Piyadasa,
mathematics. Main objective of this research 2011) and mathematicians (Ribenboim,
project is to point out that FLT can be 1991; Edwards, 1977). In these studies,
proved using elementary Mathematics and Fermat’s last theorems for = 3, = 4 and
also finding simpler and shorter proof of the theorem on the Right Triangles of
Fermat’s last theorem for = 3 than any Fermat have been proved. Fermat’s last
available proofs in the literature. In this theorem for = 4 and the theorem that the

134