Crate primal [] [src]

Simplistic and relatively unoptimised handling of basic tasks around primes:

  • checking for primality
  • enumerating primes
  • factorising numbers
  • estimating upper and lower bounds for π(n) (the number of primes below n) and pk (the kth prime)

This uses a basic Sieve of Eratosthenes to enumerate the primes up to some fixed bound (in a relatively memory efficient manner), and then allows this cached information to be used for things like enumerating the primes, and factorisation via trial division.

(Despite the name, it can sieve the primes up to 109 in about 5 seconds.)



Let's find the 10001st prime. The basic idea is to enumerate the primes and then take the 10001st in that list.

Unfortunately, Primes::sieve takes an upper bound, and it gives us no information beyond this; so we really need some way to find an upper bound to be guaranteed to include the 10001st prime. If we had an a priori number we could just use that, but we don't (for the purposes of this example, anyway). Hence, we can either try filtering with exponentially larger upper bounds until we find one that works (e.g. doubling each time), or just take a shortcut and use deeper mathematics via estimate_nth_prime.

// find our upper bound
let (_lo, hi) = slow_primes::estimate_nth_prime(10001);

// find the primes up to this upper bound
let sieve = slow_primes::Primes::sieve(hi as usize);

// (.nth is zero indexed.)
match sieve.primes().nth(10001 - 1) {
    Some(p) => println!("The 10001st prime is {}", p), // 104743
    None => unreachable!(),

Using this library

Just add the following to your Cargo.toml:

git = ""



Iterator over the primes stored in a sieve.


Stores information about primes up to some limit.


A segmented sieve that yields only a small run of primes at a time.



Returns integers (y, k) such that x = y^k with k maximised (other than for x = 0, 1, in which case y = x, k = 1).


Return Some((p, k)) if x = p^k for some prime p and k >= 1 (that is, including when x is itself a prime).


Gives estimated bounds for pn, the nth prime number, 1-indexed (i.e. p1 = 2, p2 = 3).


Returns estimated bounds for π(n), the number of primes less than or equal to n.


Test if n is prime, using the deterministic version of the Miller-Rabin test.

Type Definitions


(prime, exponent) pairs storing the prime factorisation of a number.