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 a2 + b2 = c2.

An integer solution of this equation is integers a, b and c that satisfy the equation. We all know that 32 + 42 = 9 + 16 = 25 = 52, 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 (x2 -3x+2=0 has only two integer solutions x=1 and x=2 since 0=x2 -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, an + bn = cn 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.