[FOM] Only one proof

[joeshipman@aol.com]

Almost all the important theorems of mathematics, over time, acquire 
multiple proofs.  There are many reasons for this; but I am interested 
in important theorems which, long after they are discovered, have 
"essentially" only one proof. (Only important theorems, because they 
are the ones which one would expect to be revisited enough that other 
proofs would be found.)

The best candidates I have are Dirichlet's 1837 theorem that every 
arithmetic progression with no common factor contains infinitely many 
primes, and the 1960 Feit-Thompson theorem that every group with odd 
order is solvable.

Can anyone think of other examples of comparable significance, or 
explain what is special about these theorems, or argue that one of 
these is not special because an essentially different proof than the 
original one has been found?

-- JS