-Ramanraj K
2-D and 3-D co-ordinates can be plotted with absolute certainty. However, 4-D and higher dimensions in n-dimensional spaces are not clearly defined and their co-ordinates cannot be plotted with mathematical certainty. A definition of dimensions higher than three is necessary for clarity and use in mathematics, physics and computing. This would help both man and machine to describe and visualise virtual models of the world. This is a clear definition of n-dimensional spaces with detailed diagrams. The article was first published at http://ramanraj.blogspot.com on 14th July, 2017.
The values for x, y, z, ... n that satisfy the circle equations translate to a single point in n dimensional space or nDspace. The geometry of the point when plotted as described in the clear definition of n-dimensional spaces is given here. This article was published at http://ramanraj.blogspot.com on 16th May, 2019.
Exponentiation is repeated multiplication of the base, and corresponds to symmetrical replication of the base along the axes in n-dimensional space. If b is the base, then its power or exponent n corresponds to dimensional space n extended by symmetrical replication of the base b times along the axes. This addition of exponents in nDspace is explained with illustrations.
A few rules relating to addition of exponential powers in n-dimensional space are given here. The power or exponent of an integer corresponds to the dimensional space it occupies. The identity bn + (b-1)bn = b(n+1) is critical to understanding higher dimensions, and the same is elaborated here. The basic rules governing addition of exponential powers along with elementary proofs to Pythagoras Theorem, Fermat's Last Theorem and Beal's Theorem are also discussed. This listing of basic rules is to enable visualization of integers along with their exponents in real space.
The geometry of the terms in Beal's Conjecture are drawn here using x3d. The equation in the Conjecture is derived directly from the identity Ax + (A-1)Ax = A(x+1) where A=2, and in all other cases where A>2, may be derived directly or by repeated substitution. The common prime factors are divided by their cubes to set the scale of the subcube, and the remainder is used to render the geometry of the terms. Pythagoras, Fermat, Beal, and Galois are all on the same page here. Visit page to visualize the geometry of terms in Beal's Conjecture.
The article titled "A clear definition of n-dimensional spaces" was first published at http://ramanraj.blogspot.com on 14th July, 2017. Ever since, it has been used effectively to analyze problems that are not easy to even merely view much less solve from 3D space. A list of articles in this line over the past three years are as follows:
This brief introduction to why Fermat's Last Theorem must be correct was first published on 30th April, 2019, at https://ramanraj.blogspot.com/2019/04/fairing-fermats-last-theorem.html and led to a brief four line formal proof a few months later.
This four line elementary proof for Fermat's Last Theorem first published on 12th June, 2019. Fermat's Last Theorem states it is impossible for a cube to be written as the sum of two cubes, and more generally, no three positive integers a, b, c satisfy the equation an + bn = cn for any integer value of n greater than 2. He remarked that the proof was too long to fit in the narrow margin of the book he made the note. The proof:
Pythagoras theorem states: c2 = a2 + b2
If we multiply both sides by c,
c3 = ca2 + cb2
Since the hypotenuse is greater than the sides, c > a and c > b
∴ ca2 > a3 and cb2 > b3
=> can > an and cbn > bn
=> Fermat's Last Theorem. QED.
A pictorial proof for Fermat's Last Theorem was published on 15th June, 2019.
The cubic triples and the cubic quadruples are compared in this article.
A proof for Beal's Conjecture was written at https://ramanraj.blogspot.com/2020/04/proof-for-beals-conjecture.html on 5th April, 2020.
Beal's Conjecture states that if Ax + By = Cz, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor. The equation in the Conjecture is derived directly from the identity Ax + (A-1)Ax = A(x+1) where A=2, and in all other cases where A>2, may be derived directly or by repeated substitution. It may also be noted that A, B and C must have a common prime factor or a set of common prime factors, and the greatest prime factor must be common. Further, at least one of the terms A, B and C would have either 2 or 3 as a prime factor. These properties may be used to hunt for primes.
A note on constant value 97508421 and other equivalents of 6174 for 8 digit numbers along with a form to test finalty of four digit and eight digit numbers.
The results of Kaprekar's operation on 8 digit numbers, are summarised in two tables.
Thanks to you for visiting here, and one and all who have contributed by sharing ideas through their books, communicating over social media, drawing diagrams found here, proof reading and above all for patience. Looking forward to new expansions in the light of the clear definition of n-dimensional space.