It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into two powers of the like degree; I have discovered a truly remarkable proof which this margin is too small to contain.
For the "biquadrate" (fourth power) case, Fermat's earlier assertion is sufficient to imply the later one: if two fourth powers cannot sum to a perfect square, they cannot sum to a fourth power either (since any fourth power, say w
4, is also a perfect square, namely, the square whose side measures w
2). But Fermat was asserting much more. In modern notation Fermat's assertion – known to mathematicians as Fermat's last theorem, or FLT for short – states that the equation x
n + y
n = z
n has no solution if x, y and z all are positive integers and n is a whole number greater than 2.
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