ENT

Elementary Number Theory for Integers in Rust

Algebraic definitions of primality and factorization are used, permitting checks like -127.is_prime() to return true and unique factorizations to be considered unsigned.

WIP, will probably considerably improve this over (northern hemisphere) summer of 2025.

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, Jacobi, and Kronecker 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)