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The Last Theorem of Fermat resp. Fermat-Wiles says that the equation a^n + b^n = c^n
has no solution for natural a,b,c > 0 and n > 2.

Here's a proof for sufficiently large n.
Without loss of generality be a>b, n>=b.

The idea is that at sufficiently large exponents n the term a^n + b^n lies between the n-th powers of two successive natural numbers and therefore can't be the n-th power of another natural number c.

Pascal's Triangle for (a+b)^n 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1E.g. for the case n=4:

(a+b)^4 = 1*a^4*b^0 + 4*a^3*b^1 + 6*a^2*b^2 + 4*a^1*b^3 + 1*b^4For the exponent n this means

(a+b)^n = a^n + n*a^(n-1)*b + ... + b^nEspecially for the nth power of a number that follows a^n:

(a+1)^n = a^n + n*a^(n-1) + ... + 1Since a>b, n>=b we obtain:

a^n < a^n + b^n < (a+1)^n

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