Page 147 - 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

Proof of Fermat’s Last Theorem for = without Using the Method of
Infinite Descent

Dahanayaka SD, Ekanayake EMP,Piyadasa RAD
Department of Mathematical Sciences, Wayamba University of Sri Lanka
shalikadahanayaka@gmail.com
ABSTRACT
Fermat’s last theorem (FLT) for n = 4 is proved without using the famous Method
of Infinite Descent (MID) of Fermat. Our main objective is to prove FLT for all odd primes
after proving FLT for n = 4. With respect to the proof of FLT for all indices one has to
prove the FLT for n = 4 and then to prove FLT for all odd primes. In this paper, we have
used a new simple method to prove FLT for n = 4 which can be extended for all odd
primes. First of all, we have transformed the corresponding Fermat equation to a special
4
form = ℎ + 1 where g, h are rational numbers as defined in the paper. After that we
4
have transformed the above equation to a polynomial equation of order four. Using the
properties of a polynomial equation and simple properties of a quadratic expression we
have proved the FLT for n = 4.

KEYWORDS: Fermat’s last theorem, Method of infinite descent, Odd primes

1 INTRODUCTION
1
1
4
2
= ( − ), = ( + ) and
2
4
4
4
From the equation, = + , 2 2
2
2
2
2 2
2
we can obtain the well-known primitive = (2)
Pythagorean triples of Euclid, as = This set satisfy the Pythagoras equation +
2
2
2
2
2
2, = − and = + when = where = = () , = =
2
2
2
2
2
is even and > . It is well known that ( − )/2, C = z = (a + b )/2. So, we
4
4
4
4
2
these Pythagorean triples have been used to obtain the original equations + = .
4
4
4
prove Fermat’s Last Theorem (FLT) for n =
4. This set is due to Euclid as well known 2 METHODOLOGY
and this primitive Pythagorean triples have
been used to prove Fermat’s Last Theorem We have proved the FLT for = 4
4
4

for n = 4 [1], [2]. If x is odd and another set by showing that the equation () 4 +1= () 4
of Primitive Pythagorean triples has been which can be written a. = ℎ +1 where
4
4
obtained in [3], [5] of the form = = and ℎ = , has no rational number

2
2
2
(), = − ) and = ( + )/2.
2
We can obtain Pythagorean triples, where solutions for and ℎ except = 1, ℎ = 0.
>b. From the equation
Our main objective is to prove 4 4
Fermat’s last theorem for n = 4 using a new = ℎ + 1 (3)

technique that may be used in case of all odd Where = and ℎ = ,
primes much shorter way than given in [6]. 2 2 1
If we assume that is odd and y is even, We get ( + ℎ ) ( +ℎ ) = −ℎ and
FLT for n = 4 can be stated as that; therefore − ℎ = > 0 and we get
2
2
4 4
( + )( − ) = (1) [ + 4ℎ + 6ℎ + 4ℎ − 1] = 0 (4)
2
2
2 2
4
3
3
4
2
2
From which we obtain + = Since , ℎ >0 we get
2
4
2
, − = , where > and we get 6ℎ +4ℎ − 1<0
3
2 2
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