FERMAT’S LAST THEOREM
Pierre de Fermat was a French lawyer and magistrate in
parliament of Talouse who, as a passionate amateur mathematician,
become one of the 17th century’s leading scholars. He was born on 17th
August,1601 in Beaumont- de- lomange, France in the house of
Dominique Fermat. Bieng a great mathematician he never published
any book. His contributions were very diverse which included Analytic
Geometry , Calculus(Tangents, Maxima and Minima) and Optics
(principle of least time). His son Samuel de Fermat after his death in
1665 collected all his mathematical works, edited and published them.
*THE ORIGIN OF FERMAT’S LAST THEOREM
He studied Arthmetica by Diophantus (Father of Algebra)
And wrote notes in the margin of his copy. Around 1637, He wrote one
of his famous notes, in which he claimed he had a “Truly Marvelous
proof” that no three positive integers a,b,c satisfy an +bn=cn for any
integer n>2. This statement is known as the “Fermat last Theorem”.
He also mentioned that this proof was too large to fit in the margin.
*HISTORICAL SIGNIFICANCE
This theorem remained unsolved for about 357 years.
Progress was made for some integers but no general proof was found
for centuries. Fermat himself solved the case n=4 using infinite descent.
In the 18th century Leonhard Euler proved the case of n=3. A Revolution
was brought in 19th century. First of all Sophie German developed a
powerful stratergies and proved the theorem for any prime exponents
under certain conditions. Then Ernst kumer made big progress in mid
1800s where he proved the theorem for large class of prime called
Regular prime. In the 1950-1980 connections were discovered between
fermat’s equation and Elliptic curve and moduler forms. In 1986
Kenribet proved that if a certain statement were true about the
modularity conjecture for semistable Elliptic curve the Fermat’s last
theorem would also be true. Andrew wiles solved Fermat’s last
theorem in 1994. First he gave proof in 1993 which contained small gap
which he fixed with the help from Richard taylor. Finally we get a proof
of fermat’s last theorem after 357 years. Proof spannd 129 pages which
used advanced 20th century mathematics, specifically ‘The Theory of
Elliptic curves and Moduler Forms”.
ALGEBRAIC NUMBER THEORY
( FERMAT’S LAST THEOREM :
A DRIVING FACTOR )
Algebraic number theory already existed in early form, and
fermat’s problem helped shape and accelerate its development . In mid 1800s
while Ernst kummer tried to solve fermat’s last theorem by factoring expressions
like xp + yp using complex numbers built from roots of unity (now called
cyclotomic numbers). He found out that unique factorization fails. This was the
major obstacle-it meant that the usual tools of arithmetic did not work any more.
To fix this ,kummer introduced the idea of ideal numbers(a precursor to modern
ideals). This idea was later formalized by Richard Dedekind leading to :
1. Rings of algebraic integers.
2. Ideals
3. Class groups
These are core concepts of algebraic number theory.
So, rather than creating the field from scratch fermat’s last theorem :
1. Exposed deep structural problems in number system.
2. Forced mathematicians to generalize arithmetic.
3. Motivated the creation of algebraic number theory as a systematic
discipline.
Ironically, fermat’s last theorem was finally proved in 1994 by Andrew
Wiles using tools from a completely different area- elliptic curves and
modular forms though still deeply connected to algebraic number
theory.
Fermat’s last theorem did not birth algebraic number theory it acted like
a pressure cooker which pushed mathematicians beyond classical
arithmetic, revealed hidden structures and helped transform scattered
ideas into a powerful, united field.