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: