Very Original Proofs of Two Famous Problems: “Are There Any Odd Perfect Numbers?” (Unsolved until to Date) and “Fermat’s Last Theorem: A New Proof of Theorem (Less than One and a Half Pages) and Its Generalization”

This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. Note that the second solution is very short and does not exceed one and a half pages. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.

A Brief New Proof to Fermat’s Last Theorem and Its Generalization

This article presents a brief and new solution to the problem known as the “Fermat’s Last Theorem”. It is achieved without the use of abstract algebra elements or elements from other fields of modern mathematics of the twentieth century. For this reason it can be easily understood by any mathematician or by anyone who knows basic mathematics. The important thing is that the above “theorem” is generalized. Thus, this generalization is essentially a new theorem in the field of number theory.

To the Schwarzschild Solution in General Relativity

The paper is concerned with the history of the spherically symmetric static problem solution of General Relativity found in 1916 by K. Schwarzschild [1] [2] which is interpreted in modern physics as the background of the objects referred to as Black Holes. First, the modern interpretation this solution which does not exactly coincide with original solution obtained by K. Schwarzschild is discussed. Second, the basic equations of the original Schwarzschild solution are presented in modern notations allowing us to compare existing and original solutions. Finally, a modification of the Schwarzschild approach is proposed allowing us to arrive at the exact solution of the Schwarzschild problem.

Why Krogdahl’s Flat Space-Time Cosmology Is Superior to General Relativity

This paper briefly discusses existing problems with the theory of general relativity despite remarkable accuracy in most of its applications. The primary focus is on existing problems in the field of cosmology, particularly those pertaining to expectations of global cosmic space-time curvature in the absence of observational proof. The discussion centers on Krogdahl’s recent Lorentz-invariant flat space-time cosmology and its superiority to general relativity with respect to accounting for global cosmic space-time flatness and dark energy observations. The “cosmological constant problem” is briefly addressed as a problem for general relativity with respect to particle physics and quantum field theory. Finally, two very specific validation predictions in favor of Krogdahl’s flat space-time cosmology are made with respect to ongoing studies, including the dark energy survey (DES).

Why the Road to Unification Likely Goes through Krogdahl’s Relativity

Deriving an acceptable quantum field theory of gravitation from general relativity has eluded some of the best scientific thinkers. It is gradually becoming more apparent that general relativity’s classical assumptions are simply incompatible with quantum mechanics. For instance, simultaneous certainty of the location and momentum of any moving body, regardless of size, is a fundamental feature of general relativity. And yet, special relativity and quantum mechanics (thru Heisenberg’s uncertainty) reject the very notion of simultaneity. Since special relativity is already fully integrated into quantum field theory concerning the other forces of nature, were it possible to remove the confounding smoothly curved space-time fabric of general relativity and replace it in the form of a new and improved Lorentz-invariant (flat space-time) gravitational theory, final unification might well be achievable. This brief review paper further informs the reader as to why Krogdahl’s recent Lorentz-invariant relativity model of gravitation improves on general relativity, thus providing a deeper understanding of black holes, the cosmological flatness problem and dark energy. Most importantly, since the smoothly curved space-time of general relativity may well have been the road block to unification, Krogdahl’s flat space-time model is predicted to lead to an acceptable quantum theory of gravitation (i.e., “quantum gravity”) and unification (i.e., a so-called “theory of everything”).

On the ErdÖs Distance Conjecture in Geometry

ErdÖs asks if it is possible to have n points in general position in the plane (no three on a line or four on a circle) such that for every i (1≤i≤n-1 ) there is a distance determined by the points that occur exactly i times. So far some examples have been discovered for 2≤n≤8 [1] [2]. A solution for the 8 point is provided by I. Palasti [3]. Here two other possible solutions for the 8 point case as well as all possible answers to 4 - 7 point cases are provided and finally a brief discussion on the generalization of the problem to higher dimensions is given.