One of the most fascinating features of mathematics is that many theorems are so simple to state but so difficult to prove.
A prime example is Fermat's last theorem. There was a significant amount of publicity on this theorem in 1993 when Andrew Wiles first thought he had a proof only to discover that there was an error later on. Fortunately, he was able to fix it a year later, and his proof was published in 1995. What probably contributed to the widespread publicity is that the theorem is so easy to state that many people with some mathematical education can understand. Fermat claimed he had a proof, though he wrote in his book that the margin is too narrow to contain it. Here is the theorem for those who have not seen it:
Fermat's last theorem: Given any integer 
, there are no non-zero integers 
, 
and 
such that 
.
One of my ambitious projects is to learn some elements of the proof. Apparently, it involves a deep knowledge of subjects such as elliptic curves, modular forms and Galois representations which were developed only after Fermat's time. It seems therefore unlikely that Fermat's proof of his theorem, if he had one at all, is the same proof as Wiles' proof.
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