# 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*p*(the_{k}*k*th 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 10^{10} (455052511)
on my laptop (i7-3517U).

# Using this library

Just add the following to your `Cargo.toml`

:

```
[dependencies]
primal = "0.2"
```

# Examples

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,
10^{10} will be large enough and use that, but that's a
huge overestimate (spoilers: the 10,000,000th prime is around
2×10^{8}). 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 10^{10} as the bound
instead of the more accurate estimate still only takes ~3
seconds.)

## Structs

Primes |
An iterator over all primes. |

Sieve |
A heavily optimised prime sieve. |

SievePrimes |
An iterator over the primes stored in a |

StreamingSieve |
A heavily optimised prime sieve. |

## Functions

as_perfect_power |
Returns integers |

as_prime_power |
Return |

estimate_nth_prime |
Gives estimated bounds for `n` th prime number,
1-indexed (i.e. p = 2, _{1}p = 3)._{2} |

estimate_prime_pi |
Returns estimated bounds for π( |

is_prime |
Test if |