Task — Additive primes

Write a script to find out all additive primes <= 100. Additive primes are prime numbers for which the sum of their decimal digits are also primes.

Examples

Example 1:
47, as 4 + 7 = 11, which is prime.

Analysis

We are looking for prime numbers whose digits sum to a prime. Given that I've resorted to using is_prime from Math::Prime::Util, there isn't much left to do.

One could try a number of ways of summing the digits. Perhaps the neatest (version 1) is:

$sd = 0;
$sd += $1 while $n =~ m|(\d)|g;

- a single line and reasonably comprehensible, and works for a bigint.

Another thought (version 2) was:

$n =~ s|(\d)|$1+|g;
$sd = eval($n . '0');

This inserts a plus after each digit of $n, appends a zero and evals the resulting sum.

And the arithmetic way (version 3) is:

while ($n > 0) {
    $sd += $n % 10;
    $n = int($n / 10);
}

So which is the most efficient? I tested them each on a 20-digit number, running the sum a million times, and the results were:

• Version 1 - 23 seconds
• Version 2 - 89 seconds
• Version 3 - 15 seconds.

So version 2 loses out, probably because the eval means invoking the compiler and probably also prevents any optimisation. But probably they would all be fine for any real-world task.

Script

#!/usr/bin/perl

# Peter Campbell Smith - 2022-03-30
# PWC 158 task 1

use v5.28;
use strict;
use warnings;
use utf8;

use Math::Prime::Util 'is_prime';

my ($j, $s);

# find additive primes
for $j (1 .. 100) {
    next unless is_prime($j);
    $s = $j % 10 + int($j / 10) % 10 + int($j / 100) % 10;
    say qq[$j is an additive prime whose digits sum to $s] if is_prime($s);
}

Output

2 is an additive prime whose digits sum to 2
3 is an additive prime whose digits sum to 3
5 is an additive prime whose digits sum to 5
7 is an additive prime whose digits sum to 7
11 is an additive prime whose digits sum to 2
23 is an additive prime whose digits sum to 5
29 is an additive prime whose digits sum to 11
41 is an additive prime whose digits sum to 5
43 is an additive prime whose digits sum to 7
47 is an additive prime whose digits sum to 11
61 is an additive prime whose digits sum to 7
67 is an additive prime whose digits sum to 13
83 is an additive prime whose digits sum to 11
89 is an additive prime whose digits sum to 17