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Author: heteroclinic@newmitbbs, Github Copilot


Abstract:
**Abstract**

Integer arithmetic is a cornerstone of computation, underpinning applications ranging from cryptography to distributed systems and competitive programming. However, the practical implementation of integer arithmetic often encounters challenges due to the limitations of programming languages and computational environments. This article explores the intricacies of integer arithmetic, focusing on modular arithmetic and the challenges posed by large numbers. Using a real-world problem involving unit conversions, we demonstrate how precision issues arise in JavaScript due to its reliance on 64-bit floating-point numbers, which are limited to safe integers up to \(2^{53} - 1\). We address these challenges by leveraging JavaScript's `BigInt` type, which supports arbitrary-precision integers, ensuring correctness in modular arithmetic operations. Additionally, we discuss the Unique Factorization Theorem (UFT) as a theoretical foundation for representing integers and simplifying arithmetic operations. Through examples, we showcase how UFT can be applied to represent and multiply integers efficiently. Finally, we propose a novel perspective on integer arithmetic, suggesting that integers can be represented as hashmaps of their prime factors, enabling precise and universal arithmetic operations. We conclude by inviting readers to consider the implications of this approach for real numbers and broader computational contexts.

Universal Integer Arithmetic: Bridging Theory, Practice, and Precision

Introduction
In computer science, integers are the foundation of computation. From simple arithmetic to cryptography, distributed systems, and competitive programming, integers play a pivotal role. However, the journey from mathematical theory to computational practice is fraught with challenges. This preprint explores the intricacies of integer arithmetic, the limitations of programming languages, and how we can bridge the gap between theory and practice.

Theoretical Foundations
The Unique Factorization Theorem (UFT), also known as the Fundamental Theorem of Arithmetic, states that every integer greater than 1 can be uniquely represented as a product of prime numbers, up to the order of the factors. For example, 60 = 2^2 * 3^1 * 5^1. This theorem underpins many areas of computer science, including cryptography, modular arithmetic, and number theory.

Practical Challenges
Different programming languages handle integers differently. JavaScript uses a 64-bit floating-point representation (Number), which can lead to precision issues for integers larger than 2^53 - 1. Python supports arbitrary-precision integers (int), making it ideal for handling large numbers. C++ and Java use fixed-width integer types (e.g., int, long), which can overflow if not handled carefully. These differences can lead to unexpected behavior across environments, especially when dealing with large numbers or modular arithmetic.

Showcasing the Numbers We Multiplied
Here are the numbers we worked with during the problem:

749669636 = 2^2 * 3^2 * 7^2 * 11^2 * 13^2
24357539 = 3^1 * 7^1 * 11^1 * 13^1 * 17^1
116463781 = 11^1 * 13^1 * 17^1 * 19^1
715055284 = 2^2 * 3^1 * 7^1 * 11^1 * 13^1 * 19^1

When we multiply these numbers, we can combine their prime factorizations by adding the exponents of the same primes. For example, multiplying 749669636 and 24357539 results in 2^2 * 3^3 * 7^3 * 11^3 * 13^3 * 17^1.

Another Example: Three Medium-Sized Integers
Let’s take three medium-sized integers:

120 = 2^3 * 3^1 * 5^1
210 = 2^1 * 3^1 * 5^1 * 7^1
330 = 2^1 * 3^1 * 5^1 * 11^1

Multiplying these numbers results in 2^5 * 3^3 * 5^3 * 7^1 * 11^1.

Conclusion
We believe integer arithmetic can be achieved by representing numbers as a hashmap of their prime factors, leveraging the Unique Factorization Theorem. This approach simplifies modular arithmetic, avoids overflow, and provides deeper insights into the structure of numbers. What do you think about real numbers?

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