Unsolvable Problems

A couple of years ago I popped into my local and met a friend who said "Well, what've you been doing this week?" I said "For the past few days I've been trying to solve Fermat's Last Theorem." He said: "People have been trying to do that for 4 to 5 hundred years; you don't imagine you are going to do that in a couple of days do you?"
No I didn't.
And yet....
It's one of those problems that appear so simple that you'd think anyone with a smattering of Maths could do it, until you come to try it yourself.
Another one is Goldbach's Conjecture.
Both seem capable of being solved but no one ever has.
Yet mathematicians keep receiving "proofs" of these theorems from other mathematicians, non-mathematicians, lunatics, fools.
In David Leavitt's novel "The Indian Clerk", based on real people and events, the mathematician G. H. Hardy receives a letter from an Indian clerk concerning certain mathematical formulae he has worked out. Here we go again, Hardy thought, another crank with big ideas - the letter he had received had "left a curious smell on Hardy's fingers which may have been of curry."
How could it be that a simple clerk working in India with no training in mathematics would be able to produce quality work in a very difficult field that contended for comparision with the greatest brains of the time - Hardy, Russell, Littlewood, Whithead....?
Ramanujan, the Indian clerk, came to Cambridge and proved to be an outstanding mathematician.
Fermat's Last Theorem has been proved using computers; back in the 1600's there were no computers. And Fermat had written that he possessed "a marvellous proof" but died before he demonstrated it.
That's why people keep trying to solve it in a simple way without the use of elaborate calculations on computers.
So here goes: why are the only values of n that actually work, in the equation x to the power of n plus y to the power of n equals z to the power of n, the numbers 1 and 2?
Let me see now: try 3.... No, doesn't work.... Try 4 then..... No.....
Dammit, what's Goldbach's Conjecture - that may be easier?