All,
I recently watched this youtube video talking about the sequence 2, 3, 4, 82000. 82000 is the smallest integer that when written in base 2, 3, 4 and 5 is done so only with 1s and 0s. I tried to come up with an easy way to test if a number in base 10 would be written with only 1s and 0s in another base without fully converting the number. I came up with the following (untested as I am without access to Perl at the moment).

Can you come up with a more efficient way working with the constraint that you aren't converting completely between bases? 

sub only_1_0_in_base_x {
    my ($num, $base) = @_;
    if ($num / $base != int($num / $base)) {
        $num--;
        return 0 if $num / $base != int($num / $base);
    }
    my $exponent = int(log($num) / log($base));

    my $max = 0;
    $max += $base ** $_ for 1 .. $exponent;

    return 0 if $num > $max;    

    while ($exponent) {
        my $next = $base ** $exponent;
        return 1 if $next == $num;

        $num -= $next if $num > $next;
        $max -= $next;

        return 0 if $num > $max;
    }
}

[download]

Cheers - L~R

In reply to Can It Be Written In Base X With Only 1s And 0s by Limbic~Region

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