Tag: mathematics
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On Prime Vectors and Unique Factorization of Rationals
Michael Emmerich, January 9, 2025 In this essay, we use the theory of prime vectors (see [1]) to provide an alternative proof of the unique factorization of integers into prime factors: Unlike earlier proofs [4], notably its first proof by Carl Friedrich Gauss [3], this extends unique factorization to rational numbers, and we also establish…
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Uniformly Lighting the Christmas Tree: Riesz s-Energy in Action
Uniformly Lighting the Christmas Trees: Riesz -Energy in Action Michael Emmerich, December 25th, 2024 Did you ever have the problem of how to distribute candles uniformly across your Christmas Tree? Well, here is a solution from the mathematical sciences! Using the concept of Riesz -Energy, we can optimize the placement of stars or candles on…
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Fundamental Theorem of Arithmetics: Zermelo’s proof in detail
The Fundamental Theorem of Arithmetic: Zermelo’s proof in detail Michael Emmerich, December 14th, 2024 Zermelo (1934) employs a proof by contradiction to establish the uniqueness of prime factorization for positive integers, demonstrating that the assumption of a non-unique prime factorization leads to a contradiction. Notably, his proof does not rely on Euclid’s Lemma. This essay…
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Of Autumn Leaves and Coupon Collectors
Essay by Michael Emmerich, October 11th, 2024 Imagine a peaceful autumn day where leaves gently fall, covering a patch of land. The land can be represented as a grid or matrix with distinct places, each starting uncovered. As the leaves fall, they randomly land on one of these places, gradually covering the ground. But how…
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Triangularized: Sierpinski’s Gasket and Pascal’s Triangle
Triangularized: Sierpinski’s Gasket and Pascal’s Triangle Michael Emmerich, September 28th. 2024 Introduction The Sierpinski Gasket is a fractal pattern named after the Polish mathematician Wacław Sierpiński (1882–1969). Sierpiński was a prominent figure in set theory, number theory, and topology, and he introduced several well-known fractals, including the Sierpinski Gasket (also known as the Sierpinski Triangle)…
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Playing with Primes
A Positional Notation of Integers and Rationals Based on Prime Factorization Let denote the primes, i.e. , The fundamental theorem of number theory, as proved e.g. in (Gauss, 1870), tells us that each integer can be represented as a unique product of primes. Let us represent integers as infinite lists we call prime vectors. Definition…
Michael Emmerich 