Transcript Mathematics and Art - an example put in perspective

Fermat’s Last Theorem
Dr. S. Lawrence ©2005
Who do we start from?
• Pierre de Fermat
1601-1665
civil servant all his life
mathematics his
passion and a pastime
Or do we go even further back?
• Fermat’s theorem is intriguing and exciting
because it is based on one of the simplest
and most beautiful theorems which almost
every one knows about: Pythagoras’
theorem.
• Pythagoras theorem states that …
Pythagoras’ Theorem
x 2+ y 2 = z 2
Or, in words: the square on hypotenuse is
equal to the sum of squares on two other
sides.
Fermat’s Theorem
Fermat conjectured however, that this, if
applied to cube or any other power would
not work, in other words
x n + y n = z n has no nontrivial whole
number solutions for n > 2.
You can try for yourself on few example.
Why is it a theorem and not a
conjecture?
Had Fermat been satisfied to just write it
down as we did on the previous page, it
would have remained a conjecture.
However, he wrote, in a margin of a book
he was studying at the time (famous
Arithmetica of Diophantus) that he just did
not have enough space to write the simple
proof, although he was in a possession of
one.
The exact text
Fermat wrote exactly this (translated from
Latin):
“It is impossible for a cube to be written as
a sum of two cubes or a fourth power to be
written as the sum of two fourth powers or,
in general, for any number which is a
power greater than the second to be written
as a sum of two like powers.”
“I have a truly marvelous demonstration of
this proposition which this margin is too
narrow to contain.”
Fermat’s Last Theorem
• What this actually mean in algebraic terms
is (what we said on the web page which led
you here) that although Pythagoras’
Theorem is quite valid, the equivalent for
any other power larger than 2
x n + y n = z n for n = 3, 4, 5,..
makes the equation invalid.
Hilbert’s Challenge
• In 1900 Hilbert, at the
International Congress
of Mathematicians,
challenged
mathematicians around
the world to solve
Fermat’s Last
Theorem.
Some solutions
• Few confirmations of Fermat’s
Theorem were already known
by the 1900; the quest to prove
other cases continued all the
way up to 1994 when Andrew
Wiles proved Fermat’s Last
Theorem by the Method of
Exhaustion (meaning that he
tried with all the cases that the
computer could ‘think of’ and
none have proved Fermat’s
Last Theorem to be wrong).
Nothing left to do? Not so…
• But you can still work on it…
• Think of a nicer or more ‘elegant’ proof to
confirm (or disprove!) Fermat’s Last
Theorem
• If you do, you can guarantee yourself to
have a place at the most prestigious
department of mathematics at the most
prestigious university in the world!
• Good luck!!!