Actually, the equation

is misleading. The equation should read

, where the null vector

means the dimensions don't exist for m and n at the point of intersection of the legs of the right triangle. (that is, the unit vectors

and

are orthogonal (independent); there is no component of

projected onto

and vice versa.)

The Binomial Theorem is as affirmation that Descartes was correct; that is, that there are independent dimensions, and geometric objects depend on them. John Gabriel at least (sort of) recognized the existence of two dimensions, since he did mention secants, but not curvature and continuity in functions. Grapes (and, I suspect, Wiles) don't even seem to recognize the existence of independent dimensions, effectively saying Descartes was wrong (and thus Newton and the Binomial Theorem), and that number theory is complete and consistent on a single number line (i.e., one dimension)

Some wannabe number theorists ignore Descartes altogether ( and thus geometry, and the concepts of areas, volumes, vectors, hyper-volumes, etc.) so they can concentrate on basic arithmetic operations that don't involve division, since fractions require two dimensions (a fraction is the slope of some straight line, and is a ratio that requires two independent dimensions (x,y) in the expression z = y/x - if x = 0, its dimension does not exist, so z is undefined.

(That is, if the dimension x has a metric (and as a real number, it must), then 0 can be defined as the midpoint of any length in that dimension. If there are no lengths, there is no dimension), and the fraction is undefined.

Assuming two dimensions exist, Fermat's proof boils down to:

1. For any three positive integers a,b, and c, they are all integers, or at least one of them is not.

(a) If one of them is not an integer, Fermat's Theorem doesn't apply to begin with.

2. If two dimensions exist, and at least two of the integers are on different dimensions, then there is a triangle.

3. The triangle is either a right triangle or it is not.

4. If the triangle is not a right triangle, then there will always be "rem(a,b,2)" elements in the calculation of its area, the Binomial Theorem holds, and Fermat's theorem is correct for n = 2.

5. If the triangle is a right triangle, then either it is Pythagorean or it is not.

(a)If the triangle is Pythagorean, it is the exception to Fermat's Theorem for the case n = 2, which is why Fermat restricted n to n > 2. Rotating the triangle around the point intersection of one of its legs and the hypotenuse defines a circle, whose area and circumference is a transcendental number involving

6. If the triangle is not Pythagorean, then it satisfies the Binomial Theorem for the case n=2, and rem(a,b,2) > 0

7. The Binomial Theorem is then easily extended to the case n > 2 to prove Fermat's Theorem, since rem(a,b,n) always exists and is greater than zero.

If the right triangle is considered to be one half of a rectangle (2A=ab) then the side of an equivalent square is

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