Any "modern, efficient" method of finding primes is very advanced math intensive. Applied Cryptography has a good section of prime factoring and determining if a number is likely prime. Public key encryption systems based on large prime numbers (RSA) don't actually factor the large prime numbers. They generate numbers using a technique that ensures that they are probably prime. This is a great shortcut, but won't work for claiming a mathematical record like you are proposing as the number is only probably prime (extremely high certainty that it is prime, but still not %100 guaranteed).

There are many simple prime number algorightms around, unfortunately the simple ones aren't efficient. The Sieve of Erastothenes is a classic example. I do not know of any perl modules that explicitly implement prime number type functionality. Below is a simple and extermely inefficient is_prime function.

sub is_prime{
  my $n=shift;
  return 0 if($n != int($n));
  my $m=sqrt($n);
  for(my $c=2;$c<=$m;$c++){
    return 0 if($n/$c == int($n/$c));
  }
  return 1;
}
[download]

Any way you slice it, I don't think Perl is the best language for a high performance prime number analyzer. Your best bet would be to find a language designed explicitly for doing large number math if you seriously want to go after a "largest prime number" record.