Observations on Diophantus had been written by Fermat in the margin next to Diophantus' Problem 8 in Book II, which asks 'Given a number which is a square, write it as a sum of two other squares.' Fermat's note said (in Latin) that:
On the other hand, it is impossible for a cube to be written as a sum
of two cubes or a fourth power to be written as a sum of two fourth
powers or, in general, for any number which is a power greater
than the second to be written as a sum of two like powers. I have
a truly marvellous demonstration of this proposition which this
margin is too narrow to contain.
In algebraic terms, Diophantus' problem asks for rational numbers
x, y, z satisfying the equation
x^2 + y^2 = Z^2.
This turns out to be a fairly easy task. What Fermat's marginal comment asserts is that if n is a natural number greater than 2, then the equation
x^n + y^n = z^n
has no rational solutions.
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