The following uses an iterator that returns the next highest sum of powers of the largest base. It then checks if that sum consists of just ones and zeroes in the other bases. 

use strict;
use warnings;

use Math::BigInt qw( );

# Skips over 0 and 1.
sub make_iter {
   my ($base) = @_;

   my $num = Math::BigInt->new(1);
   my @digits = 1;  # -1 represents zero.
   my @powers = Math::BigInt->new(1);

   return sub {
      if ($digits[-1] > 0) {
         push @digits, -1;
         push @powers, $powers[-1] * $base;
      }

      for (0..$#digits) {
         $digits[$_] = -$digits[$_];
         $num += $digits[$_] * $powers[$_];
         return $num if $digits[$_] > 0;
      }
   };
}

sub test {
   my ($num, $base) = @_;
   while ($num > 0) {
      my $digit = $num % $base;
      return 0 if $digit > 1;
      $num = ($num - $digit)/$base;
   }

   return 1;
}

{
   my $i = $ARGV[0] || 5;

   my $candidate_num;
   my $num;
   my $iter = make_iter($i);
CANDIDATE:
   while (1) {
      $num = $iter->();

      if (++$candidate_num == 10_000) {
         $candidate_num = 0;
         print("> $num\n");
      }

      for my $base (reverse 3..$i-1) {
         next CANDIDATE if !test($num, $base);
      }

      last;
   }

   print("$num\n");
}

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Note: If f(6) exists, it must have more than 2184 digits in base 10[Reference]. As such, I included support for large numbers.

In reply to Re: Can It Be Written In Base X With Only 1s And 0s by ikegami
in thread Can It Be Written In Base X With Only 1s And 0s by Limbic~Region

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