When Pierre de Fermat proposed that Pythagoras was unique, it became the holy grail of number theory to prove it. Mathematicians toiled for centuries trying to prove a cojecture that, at first blush, seems preposterous.
The Pythagorean Theorem
Every geometry student learns how to equate the square of the hypotenuse of a right triangle with the sum of the squares of the other two sides. Students about to take the SAT or other standardized test know to memorize the triplets 3,4,5 and 5,12,13. These are the dimensions of the right triangles with the smallest integer sides. Multiples will work too, of course. In fact, there are an infinite number of integer Pythagorean triplets.
Higher Powers
It seems like a no-brainer that the equation should work with integer solutions for powers higher than 2. For example x cubed plus y cubed = z cubed should have integer solutions, no? Well, no. Nor does the equation have integer solutions for any power higher than 2. So said Fermat. He added that he had found an ingenious proof of his conjecture, but there wasn't room to write it in the margin of his book. And, in fact, he never wrote it anywhere.
What gave credence to Fermat's claim to have found a solution was his propensity to keep proofs to himself. He frequently proclaimed a theorem and challenged others to find a proof. Most often, he was eventually shown to be correct. And it normally didn't take 350 years. However, concerning the proof of his Last Theorem, there are a number of reasons to believe that Fermat was either mistaken about his proof, or that he didn't have one.
For one thing, Fermat went out of his way to prove the theorem for the case of the power 4. If he had a general proof, there would have been no need to do this. Furthermore, many of the mathematical devices that were finally used in the proof had not been invented at the time Fermat was alive.
The Proof
Three and a half centuries after Fermat proposed his last theorem, the proof remained elusive. Finally, in 1995, Andrew Wiles provided one -- for the second time. Wiles had proclaimed the proof 3 years earlier, but on close examination by mathematicians a flaw was found. At one point, Wiles almost gave up; but he persevered and ultimately produced a proof which has now been thoroughly vetted and is not disputed.
Number Theory
It is a quirk of pure mathematics that properties appear that one would never expect. Like the fact that other than one there is only a single Munchausen number. Or that there is only a single integer that is sandwiched between a square and a cube of integers. Fermat's Last Theorem has been proved, but there is no end to the fascinating properties of numbers.
Further reading:
- Fermat's Last Theorem
- Fermat's Last Theorem, Simon Singh, 1997, Harper Centennial
Jon Plotkin - The author was a math major at Cornell and has a master's degree in meteorology from MIT.
