Question:
What's the hardest math problem you've ever done?

Replies:
It's invariably the one that I'm working on currently. This time
it's: What's the geometrical meaning of the central extension of the algebra
of diffeomorphisms of the circle?

jlu
I've been working on a problem in Number Theory off and on for almost
ten years called "the Collatz Conjecture" aka "the 3X + 1 problem".
Let f(x) be a function defined on the positive integers such that:

f(x) = x/2 if x is even

f(x) = (3*x+1)/2 if x is odd

Then the conjecture is: iterates of f(x) will eventually reach 1 for any
initial value of x. Various cash prizes have been offered for the proof
or disproof of this conjecture.

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