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Fermat's last theorem
Pierre de
Fermat
died in 1665. Today we think of
Fermat
as a number theorist, in fact as perhaps the
most famous number theorist who ever lived. It is therefore surprising to find that
Fermat
was in fact a
lawyer and only an amateur mathematician. Also surprising is the fact that he published only one
mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague's
book.
There is a statue of
Fermat and his muse
in his home town of Toulouse.
Because
Fermat
refused to publish his work, his friends feared that it would soon be forgotten unless
something was done about it. His son, Samuel undertook the task of collecting
Fermat
's letters and other
mathematical papers, comments written in books, etc. with the object of publishing his father's
mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel
written as a marginal notein his father's copy of
Diophantus
's
Arithmetica
.
Fermat's Last Theorem states that
x
n
+
y
n
=
z
n
has no nonzero integer solutions for
x
,
y
and
z
when
n
> 2.
Fermat
wrote
I have discovered a truly remarkable proof which this margin is too small to contain.
Fermat
almost certainly wrote the marginal note around 1630, when he first studied
Diophantus
's
Arithmetica
. It may well be that
Fermat
realised that his
remarkable proof
was wrong, however, since
all his other theorems were stated and restated in challenge problems that
Fermat
sent to other
mathematicians. Although the special cases of
n
= 3 and
n
= 4 were issued as challenges (and
Fermat
did
know how to prove these) the general theorem was never mentioned again by
Fermat
.
In fact in all the mathematical work left by
Fermat
there is only one proof.
Fermat
proves that
the area
of a right triangle cannot be a square.
Clearly this means that a rational triangle cannot be a rational
square. In symbols, there do not exist integers
x
,
y
,
z
with
x
2
+
y
2
=
z
2
such that
xy
/2 is a square. From
this it is easy to deduce the
n
= 4 case of
Fermat
's theorem.
It is worth noting that at this stage it remained to prove
Fermat
's Last Theorem for odd primes
n
only.
For if there were integers
x
,
y
,
z
with
x
n
+
y
n
=
z
n
then if
n
=
pq
,
(
x
q
)
p
+ (
y
q
)
p
= (
z
q
)
p
.
Euler wrote to
Goldbach
on 4 August 1753 claiming he had a proof of
Fermat
's Theorem when
n
= 3.
However his proof in
Algebra
(1770) contains a fallacy and it is far from easy to give an alternative
proof of the statement which has the fallacious proof. There is an indirect way of mending the whole
proof using arguments which appear in other proofs of
Euler
so perhaps it is not too unreasonable to
attribute the
n
= 3 case to
Euler
.
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Euler
's mistake is an interesting one, one which was to have a bearing on later developments. He needed
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 Spring '11
 re
 Number Theory, The Land, Pierre de Fermat, Fermat, Fermat's Last Theorem

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