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a^n + b^n = c^n

is solved when a,b,c being the natural numbers only for n = 2.

I would like to expand a problem:

Can anybody proove that:

a^n + b^n + c^n = d^n has a solutions a,b,c,d in the natural numbers for n = 3 and that for each higher n equation is non solveable?

Lets carry on: Can anybody proove that a^n + b^n + c^n + d^n + e^n = f^n for n = 4 the last solution exists?and for n>4 there are no solutions?

Thanks,

Robert

it would be very nice to find the rule, how many particles on the power of n you have to sum that you get the last solution of the equation in order of given n.