One of the reasons why Fermat’s Last Theorem is so difficult to prove is that it applies to an infinite number of equations: x

^{n} + y

^{n} = z

^{n}, where n is any number greater than 2. Even the advent of computers was of no help, because, although they could be employed to help perform sophisticated calculations, they could at best deal with only a finite number of equations.

Soon after the Second World War computers helped to prove the theorem for all values of n up to five hundred, then one thousand, and then ten thousand. In the 1980’s Samuel S. Wagstaff of the University of Illinois raised the limit to 25,000 and more recently mathematicians could claim that Fermat’s Last Theorem was true for all values of n up to four million. In other words, for the first four million equations mathematicians had proved that there were no numbers that fitted any of them.

This may seem to be a significant contribution toward finding a complete proof, but the standards of mathematical proofs demand absolute confidence that no numbers fit the equations for all values of n. Even though the theorem had been proven for all values n up to four million, there is no reason why it should be true for n = 4,000,001. And if in the future supercomputers proved the theorem for all values n up to one zillion, there is no reason why it should be true for n = one zillion and one. And so on ad infinitum. Infinity is unobtainable by the mere brute force of computerised number crunching.

The mathematician’s desire for an absolute proof up to infinity may seem unreasonable, but the case of Euler’s conjecture demonstrates the necessity of unequivocal truth. The 17th century Swiss mathematician Leonhard Euler claimed that there are no whole number solutions to an equation not dissimilar to Fermat’s equation:

Euler’s equation: x

^{4} + y

^{4} + z

^{4} = w

^{4}
For two hundred years nobody could prove Euler’s conjecture, but on the other hand nobody could disprove it by finding a counter-example. First manual searches and then years of computer sifting failed to find a solution. Lack of a counter-example appeared to be strong evidence in favour of the conjecture. Then in 1988 Noam Elkies of Harvard University discovered the following solution:

2,682,440

^{4} + 15,365,639

^{4} + 18,796,760

^{4} = 20,615,673

^{4}
Despite all the previous evidence, Euler’s conjecture turned out to be false. In fact Elkies proved that there are infinitely many solutions to the equation. The moral of the story is that you cannot use evidence from the first million numbers to prove absolutely a conjecture about all numbers.

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