AlgePy: Algebraic Tools in Pure Python

1. Introduction

Module Overview

AlgePy is a Python library for discrete algebra and computational number theory. It provides efficient, mathematically rigorous tools for working with number theory and algebraic structures. All algorithms are verified for complexity and correctness.

Functionality

DiscreteFunctions

• PrimalityTesting: Implements deterministic (Sieve of Eratosthenes) and probabilistic (Fermat, Miller–Rabin) methods for testing prime numbers.
• PrimeNumberTheorem: Provides functions to estimate the density and count of primes.
• Factorization: Offers algorithms to compute divisors and perform prime factorization.
• ArithmeticFunctions: Contains number-theoretic functions such as Euler’s totient, Möbius function, Liouville function, and divisor sums.

SingletonStructures

• Z: Represents elements of the ring of integers (ℤ) with overloaded arithmetic and number-theoretic checks.
• R: Models real numbers (ℝ) with standard arithmetic operations.
• Z_n & Z_mod_: Provide modular arithmetic (ℤₙ) functionality, including methods for computing orders, inverses, cyclicity, primitive roots, and the Legendre symbol.

QuadraticStructures

• C: Models complex numbers (ℂ) with standard arithmetic operations.
• Q: Represents elements of the field of rationals (ℚ) with overloaded arithmetic.
• QuadInt & QuadIntRing: Provide quadratic integer (ℤ[ω]) functionality, including methods for floor division, greatest-common-divisor (gcd), normalization, and factorization. Also provided are ring-wide functions such as fundamental and imaginary unit search.
• QuadRat & QuadRatField: Provide quadratic rational (ℚ(√d)) functionality, including true division, several types of embedding, inverses, etc.

AlgePy also integrates empirical complexity analysis to assess the performance of its algorithms.

2. Getting Started

Installation

Install AlgePy via PyPi: bash pip install algepy-tools

Or install directly from GitHub:

bash pip install git+https://github.com/sumaddury/AlgePy.git

Or install directly from source:

bash git clone https://github.com/sumaddury/AlgePy.git cd algepy pip install .

Quick Usage Example

```python from AlgePy.DiscreteFunctions import PrimalityTesting, Factorization from AlgePy.SingletonStructures import Z

Check if a number is prime using the Sieve of Eratosthenes

n = 29 print(f"{n} is prime: {PrimalityTesting.eratosthenes_primality(n)}")

Factorize a number

print("Factorization of 360:", Factorization.factorize(360))

Work with the ring of integers ℤ

a = Z(15) b = Z(10) print("a + b =", a + b) For working with quadratic systems: python from AlgePy.QuadraticStructures import QuadRatField, Q

Create a quadratic rational field Q(√2)

field = QuadRatField(2) q = field(Q(3, 1), Q(5, 1)) print("q =", q) print("q inverse =", q.inverse()) ``` CHECK: demo.ipynb for cool artwork!

3. Documentation & Analysis

Full Documentation: sumaddury.github.io/AlgePy

API Reference: Detailed API docs are automatically generated using Sphinx.

Complexity Analysis: /complexity/

4. Citation

yaml cff-version: 1.2.0 title: "AlgePy" version: "0.0.1" date-released: "2025-03-31" authors: - given-names: Sucheer family-names: Maddury repository-code: "https://github.com/sumaddury/algepy" license: "Apache-2.0"