Research Article
A Result on Odd Powers in Fermat’s Last Theorem
Issue:
Volume 10, Issue 1, April 2024
Pages:
1-6
Received:
5 December 2023
Accepted:
9 January 2024
Published:
23 January 2024
Abstract: In this work, a partial proof of Fermat’s Last Theorem (FLT) relying on elementary number theory is presented. The main result asserts that when certain natural assumptions are placed on the variables involved in the equation of the statement of FLT, then FLT holds for any prime number greater than 9, and consequently for any positive integer greater than 9. The proof of the main and supporting results is by the method of contradiction. It is first proved that if there is a prime number greater than 9 for which FLT is false under a natural assumption on the variables of the equation of FLT, then there is a set of equations that the variables must satisfy. From this set of equations, it is proved that the variables of the equation of FLT are further constrained by an additional set of equations and inequalities, which ultimately results in a contradiction. The elementary number theoretic methods employed are centered around the theory of greatest common divisors, the binomial theorem, the theory of indices, and the theory of polynomials over the ring of all integers. The algebraic operations involved are those defined on the ring of all integers, and those defined on the field of all rational numbers. The elementary order properties of the set of integers as a subset of the totally ordered field of real numbers are also applied. The cancellation and unique prime power factorization properties of the integers are taken for granted.
Abstract: In this work, a partial proof of Fermat’s Last Theorem (FLT) relying on elementary number theory is presented. The main result asserts that when certain natural assumptions are placed on the variables involved in the equation of the statement of FLT, then FLT holds for any prime number greater than 9, and consequently for any positive integer great...
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Research Article
Statistical Properties of Points Between Two Random Points
Issue:
Volume 10, Issue 1, April 2024
Pages:
7-11
Received:
17 December 2023
Accepted:
24 January 2024
Published:
5 February 2024
Abstract: Important inferences in statistics, economics and finance such as mixture distribution fitting in portfolio management are closely related to finding statistical properties of points between two random points. This problem is studied in the literature; however, accurate and fast approximations and Monte Carlo simulations are not well studied. This paper is concerned to finding these properties such as distribution function and moment generating function of points between two random points are derived. To this end, the random linear transformation technique plays important role. Also, the moment generating function is represented as expectation of random variable indexed by a Poisson variable. This note is useful to propose the Monte Carlo simulation of generating function. Two applications in mixture distribution fitting and properties of weighted averages are given. These two applications have been used in the literature for Bayesian bootstrap, change point analysis, DNA segmentations, where all theoretical results may be applied in these fields, directly. Finally, conclusions are presented.
Abstract: Important inferences in statistics, economics and finance such as mixture distribution fitting in portfolio management are closely related to finding statistical properties of points between two random points. This problem is studied in the literature; however, accurate and fast approximations and Monte Carlo simulations are not well studied. This ...
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