Escher, Concave and convex
The strategy of this proof is more
elaborate than the preceding post but retains the key idea of analyzing the
curve y = xn in terms of
the three variables a, b and c in the form of a diagram.The new proof introduces a model of
Fermat’s Conjecture based on the
implications of the anti-conjecture, the aim being to
identify inherent contradictions.
The difficulty of the problem lies in the consistency of the equations derived from the hypothesis that the
conjecture is false, the result being a series of tautologies that lead
nowhere. Although the equations that arise appear to imply the irrationality of a key
variable, it is usually the case that this condition cannot be established
conclusively. In such cases the strategy is to reject the irrationality and
follow up the implications of that decision. The situation is reminiscent of
the Escher picture shown above, where everything seems to be consistent in two
dimensions but must surely be false in a three dimensional world.
By extracting equations from the diagram, rather than just
manipulating consistent equations derived algebraically,
the inconsistencies in the assumptions are revealed and can be incorporated into a proof. This works because the geometric
picture contradicts the equations derived from the defining equation an + bn = cn.
The details of the proof are appended below:
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