Fermat's Last Theorem states for the equation: xn + yn= zn, there are no whole number solutions where x * y * z ≠ 0 and n > 2.
If x * y * z = 0, then it is easy to find a solution. For example (5)n + (0)n = (5)n.
Likewise, if we consider real numbers, then the solution is straight-forward algebra:
z = (xn + yn)(1/n).
Finally, if n = 2, then we have the Pythagorean Theorem a2 + b2 = c2. This is solveable by any Pythagorean Triple such as 3,4,5 (32 + 42 = 52) .
I think that this is the real appeal of the problem. It is easily stated and on its surface looks like it shouldn't be too difficult to resolve one way or the other.
Pierre de Fermat rarely published any of his results. He prefered to describe the problem and claim that he had found a solution. This has made the problem even more appealing: did Fermat actually have a proof?
The theorem itself became public without proof in 1670 when Fermat's son, Clement-Samuel published his father's notes. Unfortunately, Fermat was not around to explain his famous theorem because he had died in 1665. Instead, the reader was left with the famous statement of the problem:
"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 oftwo like powers."And this very mysterious statement about the proof:
"I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain." (both quotes are from Fermat's Engima)For over 350 years, this problem remained unsolved. Many of the greatest mathematicians were able to make progress on the problem including Leonhard Euler, Carl Friedrich Gauss, and Ernst Kummer but none of these great minds offered a solution.
The solution had to wait until 1995.