### The Pythagorean Theorem and Fermat's Last Theorem

Pythagoras came up with a mathematical equation that is used all the time
in architecture, construction, and measurement. The Pythagorean theorem says
that in
a right triangle (where one angle equals 90 degrees),
the sum of the squares of two sides
equals the square of the hypotenuse (the longest side).
In other words, if c is the hypotenuse, and a and b are the other two sides of a right triangle, then
a^{2} + b^{2} = c^{2}.

An **integer solution** of this equation is
integers a, b and c that satisfy the equation.
We all know that
3^{2} + 4^{2} = 9 + 16 = 25 = 5^{2},
and so we see that a=3, b=4 and c=5 is an integer solution to this equation.

Mathematicians often want to know **how many solutions**
we can find that satisfy a given equation. Sometimes there
are no solutions (5=0 has no solutions),
sometimes there is one solution (3x=6 has only one integer solution x=2),
two solutions (x^{2} -3x+2=0
has only two integer solutions x=1 and x=2 since
0=x^{2} -3x+3=(x-1)*(x-2)),
or many
solutions, and sometimes there are infinitely many solutions. Knowing
the number of solutions to an equation can have important applications,
such as applications to codes.

Fermat said that you could not find any non-zero whole number solutions to
the equation, a^{n} + b^{n} = c^{n} when n>2.
In other words, there are NO non-zero integer solutions to
this equation if n>2.

In a mathematical proof you have to write down
a line of reasoning demonstrating why there are no integer solutions.
If the proof is rigorous, then nobody can ever prove it wrong.