Prime numbers generation

Posted on September 3, 2020  (Last modified on September 4, 2020)  | 5 minutes  | 868 words
Prime numbers generation
Image from Pol Dellaiera

It all started from a book

I was reading the Open Source book from Bartosz Milewski’s ‘Category Theory for Programmers’ when I saw something about Prime numbers:

A more interesting example is a coalgebra that produces a list of primes. The trick is to use an infinite list as a carrier. Our starting seed will be the list [2..]. The next seed will be the tail of this list with all multiples of 2 removed. It’s a list of odd numbers starting with 3. In the next step, we’ll take the tail of this list and remove all multiples of 3, and so on. You might recognize the makings of the sieve of Eratosthenes.

In Python language, it would be written as such:

def naturals(n):
    yield n
    yield from naturals(n + 1)

def sieve(s):
    n = next(s)
    yield n
    yield from sieve(i for i in s if i%n!=0)

for i in sieve(naturals(2)):
    print(i) # 2, 3, 5, 7, 11, 13, 17, 19, ... 

How beautiful it is, isn’t it ?

I really like the simplicity of this algorithm, which is 2000 years old.

It’s crazy to think that such an algorithm and finding a formula to find Prime numbers is always on the wildest thoughts of every scientist in the world.

Oh you didn’t get the memo ? Let me remind it to you then… you can win 1 million dollars if you found the solution to that problem !

Finding how Prime numbers are distributed across Naturals is one of the Millenium Prize problems, if you find it too easy, you can pick another one. Cheers.

How about in PHP ?

I just did a huge refactoring in loophp/collection and I know that PHP is not the best language for this thing, but I wanted to try, you know, just to check if this is possible and compare it with the Python syntax.

At first, I had no idea on how to do it at first. I was stunned by the beauty of this algorithm and surprised to have never seen it before.

A quick look on Github has led me to some inspiration, and I started to code something and only after two days of searching, I came up with this:

<?php 

function primesGenerator(\Iterator $iterator): \Generator
{
    yield $primeNumber = $iterator->current();

    $iterator = new \CallbackFilterIterator(
        $iterator,
        fn(int $a): bool => $a % $primeNumber !== 0
    );

    $iterator->next();

    return $iterator->valid() ?
        yield from primesGenerator($iterator):
        null;
}

function integerGenerator(int $init = 1): \Generator
{
    while (true) {
        yield $init++;
    }
}

$primes = primesGenerator(integerGenerator(2));

foreach ($primes as $p) {
    var_dump($p); // 2, 3, 5, 7, 11, 13, 17, 19, ...
}

It’s not as nice as Python, but it does the job. Unfortunately, it fails quite rather quickly when XDebug is enabled.

Benchmarking

I also started to investigate how I could optimize the algorithm and made further research on it.

It turns out that this is erroneously called the “Sieve of Eratosthenes”. It should have been called the “Sieve of Trial Division”.

That algorithm as it is now, is very suboptimal, because it’s not “postponed”. Any candidate number need only be tested by Primes not above its square root. Implementing this will give an huge speedup and/because it’ll greatly minimize the stack usage.

In order to check the efficiency of this algorithm compared to a better version, I scaffolded a benchmarking tool.

As most of my work is Open Source, I quickly spawned a Github repository and created some benchmarks, using the great PHP Bench.

I implemented three different algorithms, they are almost the same with one difference:

  • Primes1: Simple Sieve of Trial Division (source)
  • Primes2: Sieve of Trial Division + Postponed (first try) (source)
  • Primes3: Sieve of Trial Division + Postponed (second try) (source)
+-------------+--------------+--------------+------------+------------+-------+
| benchmark   | subject      | mean         | mem_peak   | mem_real   | diff  |
+-------------+--------------+--------------+------------+------------+-------+
| PrimesBench | benchPrimes1 | 32,100.196μs | 2,130,696b | 4,194,304b | 1.22x |
| PrimesBench | benchPrimes2 | 31,221.970μs | 2,133,376b | 4,194,304b | 1.19x |
| PrimesBench | benchPrimes3 | 26,274.643μs | 2,156,096b | 4,194,304b | 1.00x |
+-------------+--------------+--------------+------------+------------+-------+

Primes1 is the basic algorithm based on CallbackFilterIterator where the filter callback is:

static fn (int $a): bool => (0 !== ($a % $primeNumber))

Basically this is just a sieve of trial division.

Primes2 is the same as Primes1 but the filter callback is updated to:

static fn (int $a): bool => (($primeNumber ** 2) > $a) || (0 !== ($a % $primeNumber))

There are two conditions in this callback:

  1. Any candidate number ($a) need only be tested by Primes not above its square root.
  2. The rest of the division of $a by the prime number is different from zero.

Primes3 implements a custom CallbackFilterIterator where the accept method is overridden with the same filter callback as in Primes2.

To my amazement, the algorithm Primes3 is the fastest. I still don’t get why it’s faster than Primes2, but I guess I will found out sooner or later.

It would be nice to add more algorithms and see how they behave.

If you feel like helping me and do a deep dive, feel free to clone the repo and try it out by yourself, you’ll see, it’s fun !


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