Tag: primes
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Divisibility, Co-primes, and Euler’s Totient on the Prime Vector Grid
Michael Emmerich – 29 June 2025 1. Introduction Positive integers and their fundamental building blocks, the prime numbers, exhibit a rich combinatorial structure. This essay is part of a series that aims to derive some of the intriguing properties of integers using only elementary mathematical tools, thereby making them accessible to a broader audience. We do this…
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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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RSA Cryptography: Alice and Bob
Bob and Alice sending each other messages (AI generated picture) Imagine trying to piece together a jigsaw puzzle with thousands of scattered pieces. It’s easy to make a mess, but nearly impossible to reassemble without knowing the original pattern. This is the core of a one-way function: simple to compute in one direction, but fiendishly…
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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 