PARI/GP Number Theory System

PARI/GP is a widely used computer algebra system designed for fast computations in number theory, originally developed in 1985 by Henri Cohen and a team at the University of Bordeaux in France. PARI refers to the C library, while GP is the interactive command-line interface for scripting and computation. Key features: efficient arbitrary-precision integer and rational arithmetic with fast multiplication algorithms. Algebraic number theory computations including number fields, class groups, units, Galois groups, and ideal arithmetic, making it particularly powerful for research in computational algebraic number theory. Elliptic curve computations including point counting (Schoof, SEA algorithms), modular parametrization, isogenies, torsion points, and heights, widely used in cryptographic research. Modular forms and functions including computations with modular forms, theta functions, and modular polynomials. Factorization algorithms including trial division, Pollard rho, Pollard p-1, elliptic curve method (ECM), and quadratic sieve for factoring large integers. Primality testing including deterministic and probabilistic tests with certificates. Polynomial arithmetic over various rings including finite fields, number fields, and p-adic fields. Linear algebra over integers and finite fields with Smith and Hermite normal forms. PARI C library for embedding number-theoretic computations in other applications with a clean API. GP scripting language with functional programming features, closures, and user-defined functions. High-performance vectorized operations for batch computation. Used by researchers in number theory, cryptography (RSA, elliptic curve crypto), and algebra. Cross-platform on Unix, Windows, and macOS.