Since the village idiots have forced this thread in a new forum (supposedly unrelated to physics), I will just sum up by reiterating the fact that Fermat's Theorem refers to independent integers (a,b) which must be related as vectors in Cartesian sets with independent metrics, and so are represented by ), where the unit vectors and represent independent unit metrics in each dimension.

This is represented in particular instances by the dot product of integers a and b by the null vector , which corresponds to the fact that corresponds to in Euclid's formulae in order for the expansion to be consistent for Pythagorean triples, a characterization which affirms that the triples form a right triangle, where the unit vectors are orthogonal, and so there is no projection between them.

Thus, for Pythagorean triples, the integers are not on the same number line, but each in a dimension characterized by independent orthogonal number lines. As individual elements are concerned, this represents the concept that (e.g.) , but is a vector relation where

so that

and

.

but in a single dimension.

(For the restriction to positive integers one uses both the left-hand and right-hand rule in the positive or negative quadrants of the relativistic unit circle (equal and opposite "spin", or ccw and cw Lorentz rotations) to represent the interaction term.

That is, represent the same integer on each side of the equality in a single dimension (i.e., the integer represented by the l.h.s is identical to that represented by the r.h.s. in a single dimension). (as are all arithmetic operations operations in a single dimension)

For two dimensions, then, three elements form a triangle characterized by a vector relationship.

A Pythagorean triple forms a right triangle in which all the elements are integers, so that . If a right triangle is not Pythagorean, then at least one of its elements cannot be an integer, and therefore is not the subject of Fermat's theorem.

If the triangle is a right triangle, then

(If , 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).

Then by the Binomial Theorem (well proven since the 1600's), where rem(a,b,n) > 0 by construction.

Therefore , which proves Fermat's theorem.

(Any term that "destroys" rem(a,b,n) in the Binomial Theorem must be complex, where (e.g.) (which is equivalent to taking the dot product between a and b for orthogonal vectors and ) 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.

QED

(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).

Variations in 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.

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