Elementary Number Theory for Integers in Rust
The fastest provably correct library for primality checking in the interval 0;2^64 + 2^49 that is publicly available. Algebraic definitions of primality and factorization are used, permitting checks like -127.is_prime() to return true and unique factorizations to be considered unsigned.Published as number-theory on crates.io
Currently implements these functions
- Primality
- Factorization
- GCD, Extended GCD
- Carmichael,Euler & Jordan totients
- Dedekind psi
- Liouville, and Mobius function
- Prime-counting function/nth-prime, and prime lists
- Integer sqrt/nth root
- Integer radical
- K-free
- Quadratic and Exponential residues
- Legendre symbol
- Jacobi symbol
- Smoothness checks
Additionally this library has an implementation of the previous NT functions for arbitrary-precision integers, plus some elementary arithmetic. Multiplication utilizes Karatsuba algorithm, otherwise all other arithmetic can be assumed to be naive.
- Addition/subtraction
- Multiplication
- Euclidean Division
- Conversion to and from radix-10 string
- Successor function (+1)
- SIRP-factorials {generalization of factorials}
- Conditional Interval Product (CIP factorial)
- Sqrt/nth root
- Exponentiation
- Logarithms
- Probable pseudoprime construction
Usage is fairly simple
// include NT functions
use number_theory::NumberTheory;
// include arbitrary-precision arithmetic
use number_theory::Mpz;
// Sign, generally unnecessary for ENT
//use number_theory:Sign;
let mersenne = Mpz::from_string("-127").unwrap();
assert_eq!(mersenne.is_prime(), true);
// Or for a more complex example
// check if x mod 1 === 0, trivial closure
let modulo = |x: &u64| {if x%1==0{return true} return false};
//Generate 872 factorial, using the above trivial function
// this can be just as easily reproduced as Mpz::sirp(1,872,1,0);
let mut factorial = Mpz::cip(1, 872,modulo);
// Successor function, increment by one
factorial.successor();
// 872! + 1 is in-fact a factorial prime
assert_eq!(factorial.is_prime(),true)