Doing this problem in pure Perl for small numbers is easy. Doing it in pure Perl for large numbers is hard because of round-off errors. In another language it could be easy - for instance Ruby by default uses infinite-length integers.

So one time to take my mind off of being sick I wrote a module that makes it doable, Math::FlexibleMath::Fleximal. With that you can write the following:

use Math::Fleximal;

my $number = '123456789012345678901234567890';
my $str = dec2alpha($number);
print "$str\n";
print alpha2dec($str), "\n";

# Turns a number from base 10 to base 62
sub dec2alpha {
  Math::Fleximal->new(
    shift, [0..9]
  )->change_flex(
    [0..9, 'a'..'z', 'A'..'Z']
  )->to_str();
}

# Turns a number from base 62 to base 10
sub alpha2dec {
  Math::Fleximal->new(
    shift, [0..9, 'a'..'z', 'A'..'Z']
  )->change_flex(
    [0..9]
  )->to_str();
}
__DATA__
Prints:

2AyLS9BKAMjjsWHR0
123456789012345678901234567890
[download]

Note that if you are using this to create strings to send to other people, you should consider removing various characters from the alphabet. Like the vowels, numbers that can be mistaken for being vowels (eg 3), and letters that can be mistaken for each other (eg 1 and l).

UPDATE: Corrected typo in module name. (Thanks tye.)