A GENERALIZATION OF FERMAT’S LAST THEOREMBY

ALHAJI ALKALI*AND

G. U. GARBADEPARTMENT OF MATHEMATICS

AHMADU BELLO UNIVERSITY, ZARIA A PAPER PRESENTED AT THE

45TH ANNUAL CONFERENCE OF MATHEMATICAL ASSOCIATION OF NIGERIA

(MAN), VENUE: FEDERAL COLLEGE OF EDUCATION

(TECHNICAL) GUSAU ZAMFARA STATE. 25TH – 29TH AUGUST, 2008.* Correspondence author: alkali_4real@yahoo.com

n n na b c

ABSTRACT

The Fermat last theorem states that there is no integer triple such that for

In this paper we attempt to extend this result to integer quadruples. It is shown that

is true for , and general formula for generating them is obtained.

, ,a b c

, , ,a b c dn n n na b c d

2,3n

2n

INTRODUCTION

Pierre Fermat (1601-1665) wrote a hand written comment by the side

while reading Dedekind book of Pythagoras triple that there is no

integer triple, for which for . This was

only known some years after Fermat death. There is uncertainty as to

whether Fermat had the proof of this result. This result known as

Fermat’s last theorem remain unsolved for about 500years until 1995

when Andrew Wiles in a 110-page paper was able to provide a proof

of this long standing conjecture, see [2].

, ,a b c n n na b c 2n

• for any natural number if there exit integer quadruple

for which

• general rule for obtaining all integer quadruple

for which

•Also to investigate whether there is integer quadruple

for which for where is a natural number

n

n n n na b c d

, , ,a b c d

2 2 2 2a b c d

, , ,a b c d

n n n na b c d 4n n

, , ,a b c d

Preliminary results

Proposition2.1 If is an integer quadruple, then any multiple of this is also an integer quadruple , that is for any integer n is also an integer quadruple

Proposition 2.2 For any quadruple if and are both odd integers, then and can not both be even.

Proposition 2.3 For any quadruple if and are odd then must be odd

Proposition 2.4 For any quadruple if and are even then must be even. .

, , ,a b c d

, , ,na nb nc nd

, , ,a b c d a b

c d

, ,a b , , ,a b c d c

d

, , ,a b c d , ,a b c

d

THEOREM 3.1 For any integers and , if we choose, and , then .

will satisfies then equation

Proof: Since

, ,p q r2 2 2, 2 ,a p q r b pq 2 2 2d p q r 2c qr

2 2 2 2a b c d

2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2

4 2 2 2 2 2 2 2 2 2

4 2 2 2 2 2

2 2 2 2

2

( ) (2 ) (2 )

( ( )) 4 4

2( ) ( ) 4 4

2( ) ( )

( )

a b c p q r p q q r

p q r p q q r

p q r q r p q p r

p q r q r

p q r

d

CONCLUTION This investigation was also extended to integer quadruple for which , and values are obtained that satisfy this relationship. We are now in the process of obtaining interesting results in line with this paper.However we are unable to obtain integer quadruple for which. It seen to us that there are no such integer quadruples for which for , thus extending the Fermat’s last theorem on integer triples to integer quadruples.

, , ,a b c d 3 3 3 3a b c d

, , ,a b c d4 4 4 4a b c d

, , ,a b c d n n n na b c d 4n

REFERENCE:1. John Stillwell (2002) Mathematics and its history. Springer-Verlag, New York.2. Wiles. A.(1995). Modular elliptic curves and Fermat’s

last theorem ,Ann. Of Math (2). 141(3). 443-551