Wednesday, April 2, 2014
Notes on Fermat's Last Theorem
The Fermat equation is a Diophantine equation which we may try to solve for Z as I did in FLT65. As we saw before, when p = 3, we have a quadratic equation which we may solve using the quadratic formula, and we can show relatively easily that the solutions are non-integer, proving FLT for p = 3. Also I should mention that proof for the Fermat equation when n = 4, the only non-prime that cannot be reduced to a prime ≥ 3, was not included in FLT65 because, as you know, even prior to 1965, mathematicians had proved that there are no integer solutions for exponents up to a large number, certainly past nine, which is all that is necessary for application of FLT to the work Vernon Neppe and I are doing in TDVP.
So, just for the record, since FLT was proved for all cases from n = 3 to a very large number before Wiles’ proof in 1994, and even for n = 3 and 4 much before 1965, the question of the validity of my FLT65 proof has no bearing on the validity of TDVP, which is supported by some original mathematics developed by me, including an important application of FLT to elementary particle combination.
FLT states that there are no integer solutions for any p ≥ 3. If any of the solutions to the p – 1 degree Diophantine equation (1) are integers, FLT is falsified. If we could solve this Diophantine equation generalized for all p, we would see that none of the p – 1 solutions is an integer. But this is not a trivial problem. Number ten on David Hilbert’s famous list of important unsolved mathematical problems, published in 1900, was to find a general algorithm for solving Diophantine equations.
A few other historical notes might be of interest here: The great German mathematician Johann Carl Friedrich Gauss attempted to solve this problem 100 years before Hilbert, around 1800. To Gauss, FLT was just a possible conquest as a subset of Diophantine equations in the larger quest for more general ways to solve Diophantine equations. As part of this effort, he developed what is now known in number theory as modular algebra. Appearing almost incomprehensible and complex to the uninitiated, it is really quite simple: Developed specifically for application to Diophantine equations, and thus applying only to integers, it states that an integer A is said to be “congruent” to another integer, B, modulo N, when the difference of the two integers contains N as a factor. It is written A ≡ B(Mod N), For example, 5 ≡ 3(Mod 2) simply means 5 – 3 is divisible by 2. In case you might wonder why the identity symbol is used here, recall that it is used in geometry to indicate the congruency of angles, triangles and other geometric forms.
Why would Gauss, or anyone for that matter, want to express this simple mathematical relationship in such a convoluted way? The answer lies in the fact that trinomial Diophantine equations can be reduced to binomials: For example, notice that the Pythagorean Theorem equation X2 + Y2 = Z2 may be written Z2 – X2 = Y2 and factored to obtain (Z – X)(Z + X) = Y2. For the Pythagorean integer triple (Diophantine solution) X,Y,Z = 3,4,5: (Z – X) ≡ Y(Mod Y), and (Z2 – X2) ≡ Y(Mod Y2), i.e. (25 – 9) ≡ (Mod 16). Gauss saw that this method could also be applied to the Fermat equation because it is in the same family of trinomial Diophantine polynomials as the Pythagorean Theorem equation.
By applying modular algebra to the first few prime integer exponents of Fermat’s equation, one can easily see why Gauss made the statement he did, when around 1805, he said:
“I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.”
The problem with modular algebra as an approach to solving Diophantine equations is that it only applies to binomial expressions or expressions whose terms can be re-grouped as products of binomials. Proofs that there are no integer solutions for the Fermat equation when n = 3 and 4 using this method are relatively easy to obtain because the first factor is always a binomial and the second factor can be arranged to be treated as two binomials for n = 3, and 4 binomials when n = 4. But, when n ≥ 5 the task becomes increasingly more difficult, because the number of possible combinations of terms in the second factor increases exponentially from one prime to the next, so the task of checking all of them becomes onerous very quickly. The number of terms in the second factor of the equation for any prime, p, is equal to 2p -1, i.e. 16 for p = 5, 64 for p = 7, 1,024 for p = 11, etc. And there appears to be no pattern of combinations that would allow one to check for integer solutions by groups, just like there is no apparent pattern in the occurrence of primes. Gauss went on to prove FLT for n = 3 and 4, using what are called Gaussian numbers, complex numbers which when squared are integers, also known as complex conjugates: a + bi, which are, of course, binomials handily subject to modular algebra.
A number of professional mathematicians worked on the problem of trying to find a general algorithm for solving Diophantine equations until 1970, when Yuri Matiyasevich proved that that no such general algorithm is possible within the logic of mathematics as we know it. So there is no known straight-forward way to solve Diophantine equations for all p. I can tell you, however, that application of the calculus of dimensional distinctions (CoD), developed from G. Spencer Brown’s Laws of Form by yours truly, while it does not yield the numerical values of solutions, does indicate that the Fermat equation has no integer solutions. Of course, I really don’t expect you to accept this based on my statement alone, and I don’t expect you to want to learn a whole new system of mathematics to see the logic of it for yourself. So I will try to present the essence of the CoD argument in conventional concepts below.
Calculus of Dimensional Distinctions Visualization
First, think of algebraic polynomials as distinct geometric forms. Not in the conventional sense, where x, y and z are plotted on Cartesian coordinates; instead, think of the exponent of a term as an indicator of its dimensionality. Thus terms like X, AY, (a + b)Z, etc. are linear, terms like X2, XY, aY2, etc. are planar, terms like X3, XY2, XYZ, etc. are three dimensional, and terms with variables raised to the fourth power are four-dimensional, etc. The value of X, Y, etc. in each case is a multiple of some common unit. In the case of Diophantine equations, the unit is simply the unitary integer, 1. In this scheme of things, the degree of a polynomial indicates the dimensionality of the form it describes. An unrestricted Diophantine expression is, like the ring of integers, closed but infinite. If we set a polynomial equal to a finite constant, and limit its dimensionality, it is a closed and finite distinction, like a bubble.
The Fermat expression, Zp - Xp = Y1p, where Y1 is an integer constant and p = 3, is a closed distinction that can be visualized as a spherical bubble with radius equal to Y1. And f(Z) = Z2 + XZ + Z2 is a planar form, like an ellipse cut out of a plane. The other factor, g(Z) = Z – X, can be visualized as a line. Both factors are polynomials, and their dimensionometric forms, like that of their parent polynomial, are closed and finite, limited by the value of Y1. The ellipse, with major axis smaller than the diameter of the sphere and minor axis smaller than the radius of the sphere, is enclosed within the sphere. The line, of length equal to or shorter than the minor axis of the ellipse, is enclosed within the ellipse. Now visualize a three-dimensional integer grid originating from the center of the sphere. Each node of the 3D grid is located exactly one unit from each of the six nodes nearest to it. If there are any integer solutions with Y = Y1, (X,Z) = (X1,Z1), they occur where one point on the line and two points on the ellipse coincide with three points on the surface of the sphere, and then only if those three points coincide with nodes of the integer grid. It is not too hard to see, even in this simple visualization, that this is highly unlikely, if not an outright impossibility.
We can extend this visualization to all values of p, at least conceptually, by observing that for any prime, the factorization is the same, with f(Z) a hyper-ellipsoid of degree p -1 enclosed within the hyper-sphere of the Fermat Diophantine equation with Y = Y1, and g(Z) = Z – X is a line enclosed within the ellipsoid. A hyper-dimensional integer grid is still possible to visualize because it turns out that this p-dimensional hyperspace is still Euclidean, and the Euclidean theorem for division still applies!