Fermat's Last Theorem

x^n + y^n = z^n; no integer solution.

This is Fermat's last Theorem, so named because it is the last conjecture which remained unproved among Fermat's notes. So rather, it should be called Fermat' last conjecture. For 350 years, it baffled mathematical minds, until in 1993, it was finally cracked.

"I have discovered a truly marvelous proof that it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. This margin is too narrow to contain it."

This was what Fermat claimed, on his copy of Arithmetica, and though it was a tantalizing remark, no one was sure whether it was true that he found a proof that did not require 20th century mathematics, or whether it was just one of the thousand proofs with subtle errors in them.

300 years later, inspired by media attention of the unbreakable mystery, a piece of graffiti found its way onto New York's Eight Street subway station:

x^n + y^n = z^n; no solution

I have discovered a truly remarkable proof of this, but I can't write it now because my train is coming.

Computers started to get powerful in the late 20th century, but these algorithmic machines are pretty much useless in the branch of number theory. There are infinitely many values to check for, and even if a trillion are checked, there is no guarantee that the next one would disprove the conjecture.

In the seventeenth century mathematicians showed that the following numbers are all prime:

31; 331; 3331; 33331; 333331; 3333331; 33333331

However, the next number, 333333331 turned out not to be a prime.

333333331 = 17*19607843

Euler's conjecture stated that x^4+y^4+z^4 = z^4 did not have any integer solutions. For two hundred years nobody could prove the conjecture, neither could they disprove it by finding a counter example. Then in 1988, Naom Elkies of Harvard University discovered:

2683440^4 + 15365639^4 + 18796760^4 = 20615673^4

But these two examples would be considered nothing to the next:

When Carl Gauss was 14 years old, he predicted the approximate manner in which the frequency of prime numbers would decline. The formula was rather accurate, except that it seemed to overestimate the true distribution of prime numbers. Testing for primes up to millions, billions, trillions, it just was a little too much. Thus mathematicians were tempted to believe that this would hold true for all numbers, thus giving rise to the overestimated prime conjecture...

In 1914, J.E. Littlewood proved that in a sufficiently large regime, this would no longer hold; Gauss' approximation would underestimate the number of primes. In 1955 S.Skewes showed that the underestimate would occur sometime before reaching:

10^10^10000000000000000000000000000000000

It is estimated that if one played chess with all the particles in the universe, where a move meant interchanging any two particles, then the number of possible games was roughly this number.

There was no reason that Fermat's last theorem wouldn't be as cruel as these.

Source: Simon Singh (1997). Fermat's Enigma. New York: Walker.