`primes.raku`

A short number-theory tour.

A tour through some number theory: a lazy Sieve-of-Eratosthenes sequence, factorials and Mersenne primes computed in arbitrary precision (no 64-bit ceiling), and Collatz stopping times.

The program

Edit it and press Run — it executes in your browser.

#!/usr/bin/env raku
# A short tour of number theory, leaning on Raku's arbitrary-precision
# integers (there is no 64-bit ceiling — factorials and Mersenne numbers
# grow to whatever size they need) and the lazy sequence operator `...`.

# Sieve of Eratosthenes as a lazy, self-referential sequence: each new
# prime is the next number not divisible by any prime already found.
my @primes = lazy gather {
    my @seen;
    my $n = 2;
    loop {
        unless @seen.first: $n %% * {
            @seen.push: $n;
            take $n;
        }
        $n++;
    }
}

say 'First 20 primes:';
say @primes[^20];

say '';
say 'Primes between 100 and 130:';
say @primes[^40].grep({ 100 <= $_ <= 130 });

# Arbitrary precision: factorials well past what a machine int could hold.
say '';
say 'Factorials:';
for 10, 25, 50 -> $n {
    say "  $n! = ", [*] 1 .. $n;
}

# Mersenne primes: numbers of the form 2^p - 1 that are themselves prime.
say '';
say 'Mersenne primes 2^p - 1 (p prime, p < 32):';
for @primes[^11] -> $p {
    my $m = 2 ** $p - 1;
    say "  p=$p  2^{$p}-1 = $m" if $m.is-prime;
}

# The Collatz (3n+1) sequence: the stopping time for a few seeds.
sub collatz-steps($start) {
    my $n = $start;
    my $steps = 0;
    while $n != 1 {
        $n = $n %% 2 ?? $n div 2 !! 3 * $n + 1;
        $steps++;
    }
    $steps;
}

say '';
say 'Collatz stopping times:';
for 6, 27, 97 -> $n {
    say "  $n reaches 1 in {collatz-steps($n)} steps";
}

Open in the playground ↗ Source on GitHub ↗

Output

First 20 primes:
(2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71)

Primes between 100 and 130:
(101 103 107 109 113 127)

Factorials:
  10! = 3628800
  25! = 15511210043330985984000000
  50! = 30414093201713378043612608166064768844377641568960512000000000000

Mersenne primes 2^p - 1 (p prime, p < 32):
  p=2  2^2-1 = 3
  p=3  2^3-1 = 7
  p=5  2^5-1 = 31
  p=7  2^7-1 = 127
  p=13  2^13-1 = 8191
  p=17  2^17-1 = 131071
  p=19  2^19-1 = 524287
  p=31  2^31-1 = 2147483647

Collatz stopping times:
  6 reaches 1 in 8 steps
  27 reaches 1 in 111 steps
  97 reaches 1 in 118 steps

Feature focus: lazy sequences, big Int.