It looks like a small step to go from x^2 + y^2 = z^2 to x3 + y^3 = z^3. So, following
the idea of reassembling one square around another to form a third square, can
we pull off the same trick with a cube? Can we reassemble one cube around
another to make a third? It turns out this can’t be done. The equation x^2 + y^2 =
z^2 has an infinite number of different solutions but Fermat was unable to find
even one whole number example of x^3 + y^3 = z^3. Worse was to follow, and
Leonhard Euler’s lack of findings led him to phrase the last theorem:
There is no solution in whole numbers to the equation xn + yn = zn for all
values of n higher than 2.
One way to approach the problem of proving this is to start on the low values
of n and move forward. This was the way Fermat went to work. The case n = 4
is actually simpler than n = 3 and it is likely Fermat had a proof in this case. In
the 18th and 19th centuries, Euler filled in the case n = 3, Adrien-Marie Legendre
completed the case n = 5 and Gabriel Lamé proved the case n = 7. Lamé initially
thought he had a proof of the general theorem but was unfortunately mistaken.
Ernst Kummer was a major contributor and in 1843 submitted a manuscript
claiming he had proved the theorem in general, but Dirichlet pointed out a gap in
the argument. The French Academy of Sciences offered a prize of 3000 francs for
a valid proof, eventually awarding it to Kummer for his worthy attempt. Kummer
proved the theorem for all primes less than 100 (and other values) but excluding
the irregular primes 37, 59 and 67. For example, he could not prove there were
no whole numbers which satisfied x^67 + y^67 = z^67. His failure to prove the
theorem generally opened up valuable techniques in abstract algebra. This was
perhaps a greater contribution to mathematics than settling the question itself.
Ferdinand von Lindemann, who did prove that the circle could not be squared
(see page 22) claimed to have proved the theorem in 1907 but was found to be
in error. In 1908 Paul Wolfskehl bequeathed a 100,000 marks prize to be
awarded to the first provider of a proof, a prize made available for 100 years.
Over the years, something like 5000 proofs have been submitted, checked, and
all returned to the hopefuls as being false.
The proof
While the link with Pythagoras’s theorem only applies for n = 2, the link with
geometry proved the key to its eventual proof. The connection was made with