, then the l.h.s. and r.h.s represent the same integer c = d in one dimension; i.e., either a = 0 or b = 0).
Then if a and b are independent, and do not form a Pythagorean triangle,
, where 2ab is an interaction term, which can only be eliminated if a and b are orthogonal (as in the Pythagorean right triangle).
by the Binomial Theorem (well proven since the 1600's), where rem(a,b,n) > 0 by construction.
, which proves Fermat's theorem.
(Any term that "destroys" rem(a,b,n) in the Binomial Theorem must be complex, where
(which is equivalent to taking the dot product between a and b for orthogonal vectors
) for n = 2 to retrieve the Pythagorean triple for n = 2 and all higher powers n > 2.) Note that a and b commute as scalar sums and products as scalar coefficients of the vectors in each dimension.
(The relation to physics is via the relativistic unit circle and natural logarithms, where the relativistic unit circle can be considered as an integer generator in one dimension for
, where integer generation rests on continuity instead of counting .... , so the real numbers are generated by Lorentz rotations in two dimensions rather than simply "justified" by Dedekind cuts. These are all mathematical concepts (since no physical values need be ascribed to v, c, t, or t'), since c and v are not interpreted physically but in the numerical relation
in the vector space (ct,vt') where c,v,t, and t' are (continuous) real numbers in two dimensions, where the areas are related by either irrational (e.g. geometric polygons inscribed in the circle) or transcendental numbers (concentric circles where the radius (hypotenuse) "connects" the dependent and independent variables).
are equivalent to variations in the metric tensor in the General Theory (which applies differential geometry to tensor analysis), and corresponds to "spin" in Quantum Field Theory, where positive and negative spin are introduced through the Pauli and Dirac equations (again, pure mathematics, and is also characterized by quaternions with parity (i.e.
, corresponding to "left" and "right" hand rule - that is "mirror" reflections).
This all may seem strange to village idiots, but others with a modicum of formal training will easily grasp the concepts involved; if not, pm me and I'll respond privately.