John Grieve: Dialectical analysis of Fermat's Last Theorem    
 Dialectical analysis of Fermat's Last Theorem4 comments
4 Sep 2014 @ 16:28, by John Grieve

When two unite in order to become one, then the principle of complementarity is involved. Thus X + Y = Z. This can also be seen as one splitting into two, which is fission.

With X^2 + Y^2 = Z^2 the principle is still that of complementary parts, but the form is more complex, involving squares. It is well-known that the solution of this equation involves complementarity. This is present in the angles and sides in any right-angled triangle. This complementarity has not received the recognition it deserves as the key to this solution. Since it only occurs in this one example, it has been overlooked how crucial it is.

The right-angled triangle with its complementary angles is unique. The reason why there are no solutions to higher powers of the Fermat equation is, I believe, because there are no analogous complementary structures in any other area of mathematics. You can't divide a cube into two complementary cubes because there is no underpinning equivalent to the right-angled triangle. And so on with higher powers of the equation X^N + Y^N = Z^N

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