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a^n+b^n+c^n=0
By symmetry, a solution a=b=c exists
but a, b, and c cannot all simultaneously be ==1 [mod 2] (by simple
congruences)
therefore a=b=c => a==b==c==0 [mod 2]
But if a==b==c==0 [mod 2] then there exists a smaller solution formed by
dividing out the factor of 2 from all of a,b,c
Hence, there is no smallest solution
Hence there is no integral solution
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It may amuse the reader to work out how the above reasoning doesn't
(quite) apply to the cases n=1, n=2