Peter
Peter Campbell Smith

Damm and Cyclops

Weekly challenge 177 — 8 August 2022

Week 177 - 8 Aug 2022

Task 2

Task — Palindromic prime cyclops

Write a script to generate the first 20 Palindromic Prime Cyclops Numbers. A cyclops number is a number with an odd number of digits that has a zero in the center only.

Examples


Example 1:
101

Example 2:
16061

Example 3:
31013

Analysis

Palindromic prime Cyclops numbers (PPCNs) are prime numbers with an odd count of digits with the only zero as the middle digit.

My first thought was to start at 100 and check every number for being a PPCN. This works, but isn't very efficient. My second thought was only to check numbers with an odd number of digits - so 100 to 999 and then 10000 to 99999 and so on. That was a little quicker.

However, my third and submitted algorithm looks like this:

Loop $j up from 1 to something huge. $j is going to be just the digits to the left of the central zero.

Discard any $j that contains a zero.

Now create a candidate number which is the concatenation of $j, 0 and $j reversed. So that is palindromic and has a single 0 in the right place, and all that remains is to check that it's prime. I used Math::Prime::Util to do that rather than write yet another sieve of Eratosthenes.

So far as I know, Perl has no intrinsic function to reverse a string, but this works:

join('', reverse(split('', $j)))

- that is, split the digits of the number into an array, reverse the array, and then join them together again. There may be faster methods, but using my (third) algorithm generates PPCNs up into the millions without noticeable delay.

Try it 

Try running the script with any input:



example: 17

Script


#!/usr/bin/perl

# Peter Campbell Smith - 2022-08-08
# PWC 177 task 2

use v5.28;
use utf8;
use warnings;
use Math::Prime::Util 'is_prime';

my ($j, $count, $test, $string, $k);

# loop from 1 upwards - $j is the digits before the middle 0
$k = 1;
for $j (1 .. 2 ** 31) {
    
    # no good if $j has a zero in it
    next if $j =~ m|0|;
    
    # create $j . 0 . reverse of $j
    $test = $j . '0' . join('', reverse(split('', $j)));
    
    # but is it prime?
    next unless is_prime($test);
    
    # it is
    say qq[Output: PPCN $k = $test];
    last if $k ++ == 20;
}
        

Output


Output: PPCN 1 = 101
Output: PPCN 2 = 16061
Output: PPCN 3 = 31013
Output: PPCN 4 = 35053
Output: PPCN 5 = 38083
Output: PPCN 6 = 73037
Output: PPCN 7 = 74047
Output: PPCN 8 = 91019
Output: PPCN 9 = 94049
Output: PPCN 10 = 1120211
Output: PPCN 11 = 1150511
Output: PPCN 12 = 1160611
Output: PPCN 13 = 1180811
Output: PPCN 14 = 1190911
Output: PPCN 15 = 1250521
Output: PPCN 16 = 1280821
Output: PPCN 17 = 1360631
Output: PPCN 18 = 1390931
Output: PPCN 19 = 1490941
Output: PPCN 20 = 1520251

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