The notorious mystery of Fermat’s last theorem was created by Pierre de Fermat in 1637.
Pierre Fermat (1607 – 1665) was most talked about mathematician in public life over last 358
years or more before Fermat’s last theorem got a solution from Andrew wiles. Fermat
attended University of Toulouse and later studied civil law at the University of Orléans that
helped him to lead a successful political life as a diplomat taking care of judgement etc. In
1631, he was one of the well-known and very busy government official in Toulouse and was
named as Pierre De Fermat as a mark of respect and recognition. In 1652, he was promoted to
highest chamber of criminal court. But his public life came no way to make his mathematical
ability dwarfed and in his secret private life he emerged as an enthusiast of number theory.
Till date he is remembered for Fermat’s Last Theorem (FLT) that can be stated as
x n + y n = z n
has no solution for x, y, z, n ϵ ℕ for n > 2
Pierre de Fermat was an amateur mathematician who only solved problems of mathematics in
his spare time. He was from southern France. His mathematical wisdom was largely based on
the book called Arithmetica. Arithmetica was written by Diophantus of Alexandria in the
third century AD. It was this book, where Pierre de Fermat encountered the equation
z 2 + y 2 = z 2
The equation has a close relationship with the Pythagorean Theorem. But unlike Pythagorean
theorem, where z, y and z includes the length of two sides of a right angled triangle and
hypotenuse in proportional measurement, Diophantus in Arithmetica challenged his readers
to find whole number solutions to the equation. Pierre De Fermat was in the first one who
indicate that such equation has an infinite number of whole number solutions that satisfies the
proportional measurement of length giving rise to Pythagorean triples solutions such as
3 2 + 4 2 = 5 2
5 2 + 12 2 = 13 2
133 2 + 156 2 = 205 2
……………..
and so on.
The problem was rather easy for him. Soon he graduated himself to a higher degree variant
i.e.
x 3 + y 3 = z 3
Fermat was not able to provide solution to the above equation other than the trivial one such
as
0 3 + z 3 = z 3
or he could arrive at a solution of the form
6 3 + 8 3 =9 3 -1
which is not a proper solution of the equation of x, y and z in degree 3. He wanted to probe
further and increased the power of x, y, z subsequently and finally when all his effort went
into vain to have a set of whole number solution of
x n + y n = z n where n > 2
he came to an intermediate conclusion that the equations of above form when n > 2 cannot
have a whole number solutions other than the trivial ones. This was his breakthrough thought
behind such an ordinary looking equation. The argument was that no such whole number
solution existed for set of numbers x, y, z when degree becomes more than 2. He made a note
in Latin in the margin of his copy of Diaphantus’s Arithmetica, stating that there are no
whole number solutions for any of the infinite number of equations and added ‘Cuius rei
demonstrationem mirabilem sane detexi, hanc marginis exiguitas non caperet’ meaning ‘ I
have discovered a truly marvellous proof of this, which this margin is too narrow to contain!’.
According to the above statement, Pierre De Fermat had found a proof of his earlier
statements but did not bother to write it down on some pretext taking the secret of proof away
for more than 350 years of mathematical achievement. Later after his death, Fermat’s son
Cle՜ment Samuel found his father’s copy of the Arithmetica and noticed a number of notes in
the margin of the pages in the book. Out of such noting, the above mentioned Latin sentences
were found that indulged the mathematicians of generation after generation. Fermat’s habit of
noting down gives a glimpse of his brain that could think like a logic engine leading to the
destination of proof. But the steps or path of the logic engine was never retraced. It was only
the destination that could have been found from such sparsely created proof mechanism. In
1670, Cle՜ment Samuel decided to print a new copy of Bachet’s translation of Diaphantus’s
Arithmetica with corresponding notes in the margin by Fermat for the benefit of
mathematical community. Most of the Fermat’s claims in the margin were found to be correct
and the mathematical community was delighted to have such beautiful conjectures that
generated a new interest in number theory. Except, nobody was able to prove that there were
no solution to the equation x n + y n = z n (n >2). As all other marginal notes are solved other
than this problem, the problem subsequently came to be known as Fermat’s Last Theorem,
being the unproven one. Fermat died on January 12, 1965 in Castres, France.
As the time passed, existence of a proper Fermat’s Last Theorem gained rigidity to the extent
of being infamous. The desire for the proof solely started taking its high as it caught the
imagination of many others weather from mathematics or not. Even the theorem started to
invade the personal life of many becoming the object of obsession. German Industrialist Paul
Walfskehl died in 1908 and bequeathed 100000 marks (approximately $1 million) as a
reward for anyone who could prove or disprove Fermat’s Last Theorem. Walfskehl deprived
his wife and his own family as he found solace in mathematics and not from his family. In a
way, he wanted to thank Fermat as he was fascinated with the problem. The problem became
his lifelong obsession giving him a meaning to life when he was on the verge of suicide.
Walfskehl prize further invoked public interest in Fermat’s Last Theorem (FLT). FLT started
appearing in daily conversation, in TV serials, in culture in theatre and in literature too.
The Devil and Simon Flagg’ is a short story written by Arthur Porges in 1954. The
protagonist makes a Faustian pact with Devil. Flagg could hope to save his soul by posing a
question that the Devil cannot answer and very aptly he asks for a proof of Fermat’s Last
Theorem. And Devil had no answer. Fermat’s Last Theorem made its entry in fiction (the girl
who played with fire by Stieg Larson). FLT is found in films (Bedazzled with Brendan Fraser
and Elizabeth Harley). In 1989, one episode of StarTrek: The Next Generation titled ‘The
Royale’, provides an opinion about FLT. One of its characters Captain Jean Luc Picard
describes FLT as “a puzzle we may never solve”. FLT is in play as well (Arcadia by Tom
Stoppard).
The proof or solution was actually given by Andrew Wiles at Princeton University in 1995.
With the proof, came a grand finale of all competitions for the proof of FLT. There are others
who heaved a sigh of relief that ultimately extinguish the feverish temperament of
mathematicians and others get a rest with the proof in hand. Ever since Andrew Wiles was 10
years old he encountered the simple looking, easy to understand problem in one of his library
books. The problem inspired him to appreciate the efforts put by others in this direction and
he tried to learn from their point of no return. He was obsessed with the problem for more
than three decades, out of which the last seven years he confined himself in secrecy. Finally
he proved that x n + y n = z n (n >2) has no whole number solution, though the solution in
complex plane exists. His published documentation of proof of FLT contains more than 130
pages. It showed his effort and ability to derive the logic engine to tread the path of
uncommon. However, the chain of logic is quite sophisticated to have its existence in 1637.
Wiles also used modern tools and techniques of mathematics that came into being much-
much later of 1637. Thus, though FLT has been proved successfully, certainly it is not the
proof that Fermat had in mind. In a way, Fermat still had the last laugh as his proof remained
undiscovered in spite of best efforts by all, over centuries.
A number of counterexamples were presented to show that FLT can be nullified on the basis
of these example such as
3987 12 + 4365 12 = 4472 12
The result can be verified with a calculator that can squeeze only 10 digits into its display. It
is true. But if the same result is checked with a calculator having more numbers of digits
display (not 10), then the result is
3987 12 + 4365 12 = 4472. 00000000070576171875 12
proving the earlier equation a near-miss solution of FLT. A near-miss solution is no solution
at all, keeping FLT firm is its place with crowning glory. The story of the riddle that spell
bounded the world for 358 years did not come to an end. In 2012, Shinchi Mochizhuki of
Kyoto University of Japan, while trying to prove abc conjecture successfully using his newly
found mathematical language, found an indirect proof of FLT as a corollary of a theorem. A
600 page document with a new mathematical language to be learnt is all that is required to
understand the proof of FLT by Mochizuki.
The last laugh is definitely with Fermat till today in spite of our best efforts to prove FLT.
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