Fermat's Last Theorem

I mentioned this in passing a few posts ago, but just what is it? Well, consider the following trivial sums -
7+5=12
24+6 = 30
19 +127 = 146
These all follow the pattern of a+b=c, where a, b and c are different numbers. Obviously there is an infinity of numbers which you can put in place of a, b and c. So let's take it a step further - what about a²+b²=c² that is one number squared plus another number squared equals a third number squared. There are plenty of numbers to satisfy this equation - the simplest example is 3, 4 and 5 - 3 squared (9) plus 4 squared (16) equals 5 squared (25). These are called "Pythagorean triples" but that's another story. So, a step further - a³+b³= c³ and here's the problem - nobody has ever been able to find three numbers which satisfy this equation, or indeed any such equation with a power greater than 2. This fact has been known for centuries, and the idea that such equations had no solution had become what mathematicians call a "conjecture" - that is something they are pretty sure is right, but nobody has ever been able to prove. So - to Fermat. Back in the 17th century, paper was very scarce and expensive, and so if you were reading a book and wanting to make a note about what you were reading it was commonplace to write the note in the book itself - in the margin or at the top or bottom of the page and so on. After Fermat's death, when they were going through his stuff, they came across a book which was talking about the conjecture, and Fermat had written in the margin something like "I have found a wonderful proof for this, but there isn't room for it here". Had it been anyone other than Fermat, probably not much notice would have been taken of it, but Fermat was one of the foremost mathematical minds of his time, and so this started a centuries-long search for the proof that Fermat claimed to have found. But had he? Almost certainly not - the conjecture has indeed now been proved, but only as recently as the 1990s and only then by using mathematical techniques which were not even dreamed of in Fermat's day. Along the way there have been several false proofs, which looked good until you really subjected them to close scrutiny and it was probably one of these that Fermat had found. If you want to look further into this Simon Singh has written a very good and approachable book called "Fermat's Last Theorem".