Crate primal [] [src]

primal puts raw power into prime numbers.

This crates includes

  • optimised prime sieves
  • 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 state-of-the-art cache-friendly Sieve of Eratosthenes to enumerate the primes up to some fixed bound (in a memory efficient manner), and then allows this cached information to be used for things like enumerating and counting primes.

primal takes around 2.8 seconds and less than 3MB of RAM to count the exact number of primes below 1010 (455052511) on my laptop (i7-3517U).


Using this library

Just add the following to your Cargo.toml:

primal = "0.2"


Let's find the 10001st prime. The easiest way is to enumerate the primes, and find the 10001st:

// (.nth is zero indexed.)
let p = primal::Primes::all().nth(10001 - 1).unwrap();
println!("The 10001st prime is {}", p); // 104743

This takes around 400 microseconds on my computer, which seems nice and quick, but, Primes is flexible at the cost of performance: we can make it faster. The StreamingSieve type offers a specialised nth_prime function:

let p = primal::StreamingSieve::nth_prime(10001);
println!("The 10001st prime is {}", p); // 104743

This runs in only 10 microseconds! StreamingSieve is extremely efficient and uses very little memory. It is the best way to solve this task with primal.

Since that was so easy, let's now make the problem bigger and harder: find the sum of the 100,000th, 200,000th, 300,000th, ..., 10,000,000th primes (100 in total).

We could call StreamingSieve::nth_prime repeatedly:

// the primes we want to find
let ns = (1..100 + 1).map(|x| x * 100_000).collect::<Vec<_>>();

// search and sum them up
let sum = ns.iter()
            .map(|n| primal::StreamingSieve::nth_prime(*n))
            .fold(0, |a, b| a + b);
println!("the sum is {}", sum);

This takes around 1.6s seconds to print the sum is 8795091674; not so speedy. Each call to nth_prime is individually fast (400 microseconds for 100,000 to 40 milliseconds for 10,000,000) but they add up to something bad. Every one is starting from the start and redoing work that previous calls have done... wouldn't it be nice if we could just do the computation for 10,000,000 and reuse that for the smaller ones?

The Sieve type is a wrapper around StreamingSieve that caches information, allowing repeated queries to be answered efficiently.

There's one hitch: Sieve requires a limit to know how far to sieve: we need some way to find an upper bound to be guaranteed to be at least as large as all our primes. We could guess that, say, 1010 will be large enough and use that, but that's a huge overestimate (spoilers: the 10,000,000th prime is around 2×108). We could also try filtering with exponentially larger upper bounds until we find one that works (e.g. doubling each time), or, we could just take a shortcut and use deeper mathematics via estimate_nth_prime.

// the primes we want to find
let ns = (1..100 + 1).map(|x| x * 100_000).collect::<Vec<_>>();

// find our upper bound
let (_lo, hi) = primal::estimate_nth_prime(10_000_000);

// find the primes up to this upper bound
let sieve = primal::Sieve::new(hi as usize);

// now we can efficiently sum them up
let sum = ns.iter()
            .map(|n| sieve.nth_prime(*n))
            .fold(0, |a, b| a + b);
println!("the sum is {}", sum);

This takes around 40 milliseconds, and gives the same output: much better!

(By the way, the version using 1010 as the bound instead of the more accurate estimate still only takes ~3 seconds.)



An iterator over all primes.


A heavily optimised prime sieve.


An iterator over the primes stored in a Sieve instance.


A heavily optimised prime sieve.



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.