"Fermat's Last Theorem" Disproved?
in [Math]
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From: cjcountess on 24 Jul 2010 15:54 Just what does it mean to square something? Well here are at least 3 ways 1. Geometrically, if you take a line of 1 inch in the linear direction, and multiply it by a line of 1 inch in the 90 degree angular direction, you get a square inch 2. In equation F=mv^2, the force or (F), is equal to the mass or (m) times the velocity or (v) ^2 or squared. This means that, each time velocity doubles, force quadruples. 3. And finally, a velocity vector in the linear direction, x an equal and 90 degree angular velocity vector = v^2 = circular motion as a balance of centrifugal centripetal forces, and can also use equation F=mv^2, F=mv^2/r or F=mv/r^2 The formula F=mv^2 means that each time velocity doubles force quadruples, the term v^2 or velocity squared can also mean that a velocity vector in the linear direction x a velocity vector in 90 degree angular direction leads to circular motions as a balance of centrifugal centripetal forces. and F=mv^2/r can be used for this motion as in orbital motions also, and a unite length squared, can mean 1 unite length in linear direction x 1 unite length in 90 degree angular direction to create a square of that unite. How do all these related? Well, if we drop a mass from the upper atmosphere toward the earth, the formula (F=mv^2/r), dictates that the force will quadruple, each time the velocity doubles as the radius between mass and earth shortens as mass approaches the earth, which it does at 32fps^2, and if allowed to fall to the center unencumbered, it will spin at a constant v^2. Likewise, if an object is shot into orbit from earth, if it reaches a centrifugal velocity at 90 degree angle to and equal to the centripetal force of the gravity of the earth, without overshooting and reaching an escape velocity, it will orbit the earth. An object moving across the earth or falling toward the earth will increase in speed and corresponding relative mass, kinetic energy, at 32ftps^2, and if allowed to fall straight through the earth until it reaches the center will either bounce back and forth like a pendulum until it comes to a spinning stability at the earths center. As such v^2 goes from an increasing linear motion with corresponding increase relative mass and kinetic energy, to a circular motion just as orbiting the earth orbiting the earth or spinning at center of earth. And both linear and circular motion can use the formula F=mv^2, F=mv/ r^2, or F=mv^2/r. And so liner motion and circular motion converge through v^2. And as I pointed out before, so does spherical motion, as a geometrical interpretation of E=mc^2 reveals that a photon that begins as a relatively straight line bends into a wave as it is squeezed against the light barrier becoming more particle like as it gains mass energy and momentum and at c^2 attains rest mass through circular and or spherical motion. c^2 is also a measure of the energy in all 3D spacial objects, even on macro level. To me even if not to others, it is obvious that the same process one uses to square a unite length, is exactly the same as that used to extend that same square into 3D space, and as such, a cube can be viewed geometrically, as the square of a square. Posters have pointed out to me that the square of a square is a forth power and not a cube. That does relate to F=mv^2,as the force increase 4 x each time velocity doubles. But it does not relate well in my mind to geometrical 3D space of which we know that E=mc^2 is a measure of the energy of. Furthermore, what is the volume of a 4D object, unless as I said, the time dimension is involved. And, is this square to the forth power more that a cubic space in volume? If the square of a square = 4 squares would that obey the Pythagorean theorem? If we take a 90 degree triangle of length 3 for one right angle , 4 for the other and 5 for the hypotenuse and square each side we get a 3 inch square a 4 inch square and a 5 inch square and if we apply the Pythagorean theorem we get 9 + 16 =25, which satisfies the theorem. If we furthermore square the sides by multiplying them by 4 we get 36 + 64 = 100. And so the theorem is extended to powers beyond that of 2 to 4 and beyond. One poster said that the Fermat theorem only referred to integers, but we are discussing 1D geometrical unite lengths, as the sides of a right triangle, and that is geometry. That said, the Fermat theorem does break down at the square of the square, weather it be the increase from 1 to 4 squares, or the extending of a square into 3D space as a cube. If we take the equation E=hf/c^2 which pertains to the energy of photons, it states that energy equals Plancks constant or (h), time the frequency of (f) divided by c^2. Furthermore that said energy increases with the square of the frequency which is 4 time each time frequency doubles. Yet the energy doesn't reach the level of c^2 to create matter or rest mass, until the end of the EM spectrum, where as deBroglie discovered, E=hf=mc^2. And so the doubling of the frequency over and over again which is like doubling of speed as the cycles of the photon and its corresponding quadrupling of the mass/ energy continues and does not reach c^2 even as the frequency increased a thousand fold until the end of the EM spectrum. The quadrupling of the mass/energy as the frequency doubles, may be analogous to the square of the square as it increases 4 times but still does not equal the square of the square as a cube. In other words, just as it takes more than 4 lines stacked upon each other to make a square of that unite length, it also takes more than 4 squares stacked upon each other to square a square, geometrically to make a cube. The cube in this sense is analogous to c^2 at end of EM spectrum, which is more than the square of the square as an increase of 4 squares, which would be analogous to an increase in frequency. Well how is it that c^2 leads to 3D material world? In plain geometry we can extend a 2D object into 3D space by repeating the same process we did to extend a 1D line into 2D, we square it. This requires raising the object the same length in all 3 dimensions, which requires adding more material to the square to make a cube. But on quantum level c^2 as c in circular motion, as a 2 D circle, is extended into 3D space, by folding itself in half, and making two rotations at right angle, to complete one wave cycle. This creates a 3D standing spherical wave, out of a 2 D circle, by folding, and without adding any more material to it. Still no matter how you slice it, the 3D world is composed of energy squared and as such so is a cube. Furthermore, to create a 3D cube from a 2D surface, one has to do the same exact thing one does to extend a 1D line to a 2D surface, square it. And although the cube may not follow the Pythagorean or Fermat theorems, it is not because the cube is not the result of squaring. Conrad J Countess
From: spudnik on 25 Jul 2010 20:38 <deleta impleta> thus: 3 choices, 2 choices, 1 choices (3?, or "three summorial" .-) yeah, direction cosines are nice & homogenous, but why not stay with vectors (quaternions' inner & outer products) ?? thus: IFF probably is "if & only if," that is to say, Liebniz's neccesity & sufficiency, used in literate manner! > Iff ... then ... --les ducs d'oil! http://tarpley.net --Stop BP's cap&trade looting! http://wlym.com
From: Frederick Williams on 27 Jul 2010 12:17 cjcountess wrote: > > Just what does it mean to square something? If you mean square a number, it just means multiply it by itself. > Well here are at least 3 ways > > [nonsense snipped] -- I can't go on, I'll go on.
From: cjcountess on 28 Jul 2010 10:44 Before final judgment is passed, has anyone considered this. Considering this Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. from http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem But it may just as well be true that "no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value period", if the unites involved are not geometrical unites lengths related to Pythagorean Theorem and sides of a right triangle. And unite lengths are just as different from abstract non dimensional integers as 1x1=1 is to 1D geometric line in horizontal direction x 1D geometrical line in vertical direction = a 2D square unit, and as such "Fermat's Last Theorem", can not be based on integers, with n > 2 or otherwise, and is therefore ? As integers can be separate from geometry, but converge with geometry at point where "a^2 + b^2 = c^2" can we really say that at this point of convergence that it is still separate from geometry, and would therefore still be the case that "a^2 + b^2 = c^2", if not for this convergence with geometry that may or may not be separated from it. In other words can "a^2 + b^2 = c^2" hold true without dimensional geometry and therefore are they truelly dimensionless integers? Conrad J Countess
From: Kermit Rose on 28 Jul 2010 12:15
"cjcountess" <cjcount...(a)yahoo.com> wrote on Jul 14, 2010 5:18 PM > Has anyone taken two cubes, created from two equal and > right angular lines of triangle, and one made from cube > of hypotenuse, and measured their volumes, to see if > the ?c? cube equals the ?a? cube + the ?b? cube? > Has this ever been done that anyone knows of ? > That would be the proof. In fact this has been done many times. 3^3 + 4^3 + 5^3 = 6^3. 027 064 125 --- 216 = 6^3 Since in a cube there are 3 diagonals, there needs to be three terms in the sum of cubes, not 2. Maybe you have hit on another approach to prove in fact that Fermat's last theorem is true. Kermit Rose. |