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/*!
A basic library for finding primes, providing a basic Iterator over all primes. It is not as fast as
`slow_primes`, but it is meant to be easy to use!
The simplest usage is simply to create an `Iterator`:
```
use primes::{Sieve, PrimeSet};
let mut pset = Sieve::new();
for (ix, n) in pset.iter().enumerate().take(10) {
println!("Prime {}: {}", ix, n);
}
```
This library provides methods for generating primes, testing whether a number is prime, and
factorizing numbers. Most methods generate primes lazily, so only enough primes will be generated
for the given test, and primes are cached for later use.
[*Source*](https://github.com/wackywendell/primes)
# Example: Find the first prime after 1 million
```
use primes::{Sieve, PrimeSet};
let mut pset = Sieve::new();
let (ix, n) = pset.find(1_000_000);
println!("Prime {}: {}", ix, n);
```
# Example: Find the first ten primes *after* the thousandth prime
```
use primes::{Sieve, PrimeSet};
let mut pset = Sieve::new();
for (ix, n) in pset.iter().enumerate().skip(1_000).take(10) {
println!("Prime {}: {}", ix, n);
}
```
# Example: Find the first prime greater than 1000
```
use primes::{Sieve, PrimeSet};
let mut pset = Sieve::new();
let (ix, n) = pset.find(1_000);
println!("The first prime after 1000 is the {}th prime: {}", ix, n);
assert_eq!(pset.find(n), (ix, n));
```
For more info on use, see `PrimeSet`, a class which encapsulates most of the functionality and has
multiple methods for iterating over primes.
This also provides a few functions unconnected to `PrimeSet`, which will be faster for the first
case, but slower in the long term as they do not use any caching of primes.
*/
#![doc(html_root_url = "https://wackywendell.github.io/primes/")]
use std::cmp::Ordering::{Equal, Greater, Less};
use std::cmp::Reverse;
use std::collections::BinaryHeap;
use std::ops::Index;
use std::slice;
pub trait PrimeSetBasics {
/// Finds one more prime, and adds it to the list
fn expand(&mut self);
/// Return all primes found so far as a slice
fn list(&self) -> &[u64];
}
/**
A prime generator, using the Trial Division method.
Create with `let mut pset = TrialDivision::new()`, and then use `pset.iter()` to iterate over all
primes.
**/
#[derive(Clone)]
pub struct TrialDivision {
lst: Vec<u64>,
}
const WHEEL30: [u64; 8] = [1, 7, 11, 13, 17, 19, 23, 29];
#[derive(Copy, Clone)]
struct Wheel30 {
base: u64,
ix: usize,
}
impl Wheel30 {
pub fn next(&mut self) -> u64 {
let value = self.base + WHEEL30[self.ix];
self.ix += 1;
if self.ix >= WHEEL30.len() {
self.ix = 0;
self.base += 30;
}
value
}
}
impl Default for Wheel30 {
fn default() -> Self {
Wheel30 { base: 0, ix: 1 }
}
}
/**
A prime generator, using the Sieve of Eratosthenes method. This is asymptotically more efficient
than the Trial Division method, but slower earlier on.
Create with `let mut pset = Sieve::new()`, and then use `pset.iter()` to iterate over all primes.
**/
#[derive(Clone)]
pub struct Sieve {
primes: Vec<u64>,
wheel: Wheel30,
// Keys are composites, values are prime factors.
//
// Every prime is in here once.
//
// Each entry corresponds to the last composite "crossed off" by the given prime,
// not including any composite less than the values in 'primes'.
sieve: BinaryHeap<Reverse<(u64, u64)>>,
}
/// An iterator over generated primes. Created by `PrimeSet::iter` or
/// `PrimeSet::generator`
pub struct PrimeSetIter<'a, P: PrimeSet> {
p: &'a mut P,
n: usize,
expand: bool,
}
impl TrialDivision {
/// A new prime generator, primed with 2 and 3
pub fn new() -> TrialDivision {
TrialDivision { lst: vec![2, 3] }
}
}
impl Default for TrialDivision {
fn default() -> Self {
Self::new()
}
}
impl PrimeSetBasics for TrialDivision {
/// Finds one more prime, and adds it to the list
fn expand(&mut self) {
let mut l: u64 = self.lst.last().unwrap() + 2;
let mut remainder = 0;
loop {
for &n in &self.lst {
remainder = l % n;
if remainder == 0 || n * n > l {
break;
}
}
if remainder != 0 {
self.lst.push(l);
break;
};
l += 2;
}
}
/// Return all primes found so far as a slice
fn list(&self) -> &[u64] {
&self.lst[..]
}
}
impl Default for Sieve {
fn default() -> Self {
Self::new()
}
}
impl Sieve {
/// A new prime generator, primed with 2 and 3
pub fn new() -> Sieve {
Sieve {
primes: vec![2, 3, 5],
sieve: BinaryHeap::new(),
wheel: Wheel30 { base: 0, ix: 1 },
}
}
// insert a prime and its composite. If the composite is already occupied, we'll increase
// the composite by prime and put it there, repeating as necessary.
fn insert(&mut self, prime: u64, composite: u64) {
self.sieve.push(Reverse((composite, prime)));
}
}
impl PrimeSetBasics for Sieve {
/// Finds one more prime, and adds it to the list
fn expand(&mut self) {
let mut nextp = self.wheel.next();
loop {
let (composite, factor) = match self.sieve.peek() {
None => {
self.insert(nextp, nextp * nextp);
self.primes.push(nextp);
return;
}
Some(&Reverse(v)) => v,
};
match composite.cmp(&nextp) {
Less => {
let _ = self.sieve.pop();
self.insert(factor, composite + 2 * factor);
}
Equal => {
let _ = self.sieve.pop();
self.insert(factor, composite + 2 * factor);
// 'nextp' isn't prime, so move to one that might be
nextp = self.wheel.next();
}
Greater => {
// nextp is prime!
self.insert(nextp, nextp * nextp);
self.primes.push(nextp);
return;
}
}
}
}
/// Return all primes found so far as a slice
fn list(&self) -> &[u64] {
&self.primes[..]
}
}
pub trait PrimeSet: PrimeSetBasics + Sized {
/// Number of primes found so far
fn len(&self) -> usize {
self.list().len()
}
fn is_empty(&self) -> bool {
self.list().is_empty()
}
/// Iterator over all primes not yet found
fn generator(&mut self) -> PrimeSetIter<Self> {
let myn = self.len();
PrimeSetIter {
p: self,
n: myn,
expand: true,
}
}
/// Iterator over all primes, starting with 2. If you don't care about the "state" of the
/// `PrimeSet`, this is what you want!
fn iter(&mut self) -> PrimeSetIter<Self> {
PrimeSetIter {
p: self,
n: 0,
expand: true,
}
}
/// Iterator over just the primes found so far
fn iter_vec(&self) -> slice::Iter<u64> {
self.list().iter()
}
/// Find the next largest prime from a number
///
/// Returns `(idx, prime)`
///
/// Note that if `n` is prime, then the output will be `(idx, n)`
fn find(&mut self, n: u64) -> (usize, u64) {
while n > *(self.list().last().unwrap_or(&0)) {
self.expand();
}
self.find_vec(n).unwrap()
}
/// Check if a number is prime
///
/// Note that this only requires primes up to `n.sqrt()` to be generated, and will generate
/// them as necessary on its own.
fn is_prime(&mut self, n: u64) -> bool {
if n <= 1 {
return false;
}
if n == 2 {
return true;
} // otherwise we get 2 % 2 == 0!
for m in self.iter() {
if n % m == 0 {
return false;
};
if m * m > n {
return true;
};
}
unreachable!("This iterator should not be empty.");
}
/// Find the next largest prime from a number, if it is within the already-found list
///
/// Returns `(idx, prime)`
///
/// Note that if `n` is prime, then the output will be `(idx, n)`
fn find_vec(&self, n: u64) -> Option<(usize, u64)> {
if n > *(self.list().last().unwrap_or(&0)) {
return None;
}
let mut base: usize = 0;
let mut lim: usize = self.len();
// Binary search algorithm
while lim != 0 {
let ix = base + (lim >> 1);
match self.list()[ix].cmp(&n) {
Equal => return Some((ix, self.list()[ix])),
Less => {
base = ix + 1;
lim -= 1;
}
Greater => (),
}
lim >>= 1;
}
Some((base, self.list()[base]))
}
/// Get the nth prime, even if we haven't yet found it
fn get(&mut self, index: usize) -> u64 {
for _ in 0..(index as isize) + 1 - (self.list().len() as isize) {
self.expand();
}
self.list()[index]
}
/// Get the prime factors of a number, starting from 2, including repeats
fn prime_factors(&mut self, n: u64) -> Vec<u64> {
if n == 1 {
return Vec::new();
}
let mut curn = n;
let mut lst: Vec<u64> = Vec::new();
for p in self.iter() {
while curn % p == 0 {
lst.push(p);
curn /= p;
if curn == 1 {
return lst;
}
}
if p * p > curn {
lst.push(curn);
return lst;
}
}
unreachable!("This should be unreachable.");
}
}
impl<P: PrimeSetBasics> PrimeSet for P {}
impl Index<usize> for TrialDivision {
type Output = u64;
fn index(&self, index: usize) -> &u64 {
&self.list()[index]
}
}
impl<'a, P: PrimeSet> Iterator for PrimeSetIter<'a, P> {
type Item = u64;
fn next(&mut self) -> Option<u64> {
while self.n >= self.p.list().len() {
if self.expand {
self.p.expand()
} else {
return None;
}
}
self.n += 1;
let m = self.p.list()[self.n - 1];
Some(m)
}
}
/// Find the first factor (other than 1) of a number
fn firstfac(x: u64) -> u64 {
if x % 2 == 0 {
return 2;
};
// TODO: return to step_by
// for n in (3..).step_by(2).take_while(|m| m*m <= x) {
for n in (1..).map(|m| 2 * m + 1).take_while(|m| m * m <= x) {
if x % n == 0 {
return n;
};
}
// No factor found. It must be prime.
x
}
/// Find all prime factors of a number
/// Does not use a `PrimeSet`, but simply counts upwards
pub fn factors(x: u64) -> Vec<u64> {
if x <= 1 {
return vec![];
};
let mut lst: Vec<u64> = Vec::new();
let mut curn = x;
loop {
let m = firstfac(curn);
lst.push(m);
if m == curn {
break;
} else {
curn /= m
};
}
lst
}
/// Find all unique prime factors of a number
pub fn factors_uniq(x: u64) -> Vec<u64> {
if x <= 1 {
return vec![];
};
let mut lst: Vec<u64> = Vec::new();
let mut curn = x;
loop {
let m = firstfac(curn);
lst.push(m);
if curn == m {
break;
}
while curn % m == 0 {
curn /= m;
}
if curn == 1 {
break;
}
}
lst
}
/// Test whether a number is prime. Checks every odd number up to `sqrt(n)`.
pub fn is_prime(n: u64) -> bool {
if n <= 1 {
return false;
}
firstfac(n) == n
}