The holy grail of mathematics.
Background of Fermat’s Last Theorem:
Let us first look at positive Integral Solutions of
some basic equations. For example let us take an equation a+b=c. How many
positive Integral Solutions does this equation have? Obviously Infinite. We can
see that 2+3= 5 or 5+10= 15 and like that infinite solutions.
The next example is a2+b2=c2.
How many positive Integral Solutions does this equation have? All Pythagorean
triplets like 3-4-5, 5-12-13, 8-15-17, 7-24-25… and their multiples like 6-8-10
etc; all are solutions of this equation. So, this equation too has infinite
solutions. This particular equation a2+b2=c2
and its solutions are known to mankind for the past two thousand years.
Such polynomial equations like a2+b2=c2
in which the solutions must be positive integers are known as Diophantine
equations. Their name derives from the 3rd-century Alexandrian mathematician,
Diophantus, who developed methods for their solutions. Diophantus's major work
is the Arithmetica which was written in the third century and remained
the source for such equations for next several centuries.
Pierre de Fermat thought about the next higher level Diophantine equations like:
a3+b3=c3
a4+b4=c4
a5+b5=c5
…………………..
an+bn=cn
He spent several years on the above equations and could not find any integral solutions for the above equations. Finally he was convinced that
an+bn≠cn
Where a,b, c and n are natural Numbers and n>2.
And the above is what is known as Fermat’s Last Theorem.
Pierre de Fermat wrote many theorems on number Theory
which all were proved subsequently except the one as mentioned above. However,
around 1637, Fermat wrote his above–mentioned most famous Theorem in the margin
of his copy of the Arithmetica next to Diophantus' sum-of-squares
problem:
“It is impossible to separate a cube into two
cubes, or a fourth power into two fourth powers, or in general, any power
higher than the second, into two like powers. I have discovered a truly
marvelous proof of this, which this margin is too narrow to contain”.
Those were the most mysterious words ever written by
Pierre de Fermat. Decades and centuries went by but nobody could locate the
proof written by him. It was lost for ever. It sort of became a big
mathematical challenge to prove the theorem. Mathematicians started considering
it as a holy grail of Mathematics. Hundreds of Mathematicians including Euler,
Gauss, Newton and their worthy successors worked on the problem but could never
prove the theorem in the next three and a half centuries till 1995.Many
Mathematicians devoted their lives just on this theorem but could not succeed.
No successful proof was published until 1995 despite the efforts of countless
mathematicians during the 358 intervening years.
That abnormally long time-span of more than 350 years
accentuates the significance of this Conjecture- Fermat’s last theorem. It is
hard to conceive of any problem, so simply and clearly stated that could have
withstood the test of advancing knowledge for so long.
Let us consider the progress in physics, chemistry,
biology, medicine and engineering that have taken place since the seventeenth
century. All the subjects have changed beyond recognition. The Physics became
quantum Physics. Medicine metamorphosed into genetics. Modern Chemistry makes
the chemistry of 17th century look like alchemy. There is no
comparison of the primitive technology of the seventeenth century to today’s
space age. But the Fermat’s last theorem (FLT) could not be conquered
till 1995.
Mount Everest could be conquered in two to three
decades but the FLT took more than 350 yrs. Fermat’s Last Theorem was the
Himalayan peak of number theory. And Fermat was the father of modern number
theory.
Prizes Declared on the Proof of Fermat’s last theorem:
In 1816 and in
1850, the French Academy of Sciences offered a prize for a general proof of
Fermat's Last Theorem. Another prize was offered in 1883 by the Academy of
Brussels.
In 1908, the German
industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 marks
to the Göttingen Academy of Sciences to be offered as a prize for a complete
proof of Fermat's Last Theorem.
Finally Andrew Wiles a British Mathematician proves the Theorem in 1995:
Sir Andrew John Wiles, a British mathematician and a
Royal Society Research Professor at Oxford University, specializing in number
theory ultimately proved the Fermat’s last theorem in 1995 with the help of
Computers and latest research in the field of Number Theory.
Wiles came to know about Fermat’s Last Theorem when he
was 10 years old. He was Surprised by the fact that the statement of the
theorem was so easy that he at the age of 10 could understand it.He aspired to
be the first person to prove it. However, he soon realized that his knowledge
of mathematics was too inadequate, so he abandoned his childhood dream, until
1986, when he heard that Ribet, a famous Mathematician, had established a link
between Fermat's Last Theorem and the Taniyama–Shimura conjecture about
elliptical curves and modular functions which are intricately linked to each
other.
Based on successive progress of the previous few years
of renowned Mathematicians like Gerhard Frey, Jean-Pierre Serre and Ken Ribet;
he dedicated all of his research time to this problem in relative secrecy. In
1993, he presented his proof to the public for the first time at a conference
in Cambridge. However certain errors were noticed by Mathematicians in the
Proof.
Wiles tried to correct the mistakes but found out that
the gap he had made was significant. The essential idea bypassing, rather than
closing this gap, came to him in September 1994. Together with his former
student Richard Taylor, he published a second paper which circumvented the gap
and thus completed the proof. Both papers were published in 1995 in a special
volume of the Annals of Mathematics.
Wiles’ proof is well over 100 pages long, and involves
some of the most advanced Mathematics of its time, and so the question does
linger, “Is there a ‘simple’ proof of the Theorem as originally claimed by
Fermat?”
