TL;DR

On with TASK #2 from The Weekly Challenge #157. Enjoy!

# The challenge

You are given a number $n > 3. Write a script to find out if the given number is a Brazilian Number. A positive integer number N has at least one natural number B where 1 < B < N-1 where the representation of N in base B has same digits. Example 1: Input:$n = 7
Output: 1

Since 7 in base 2 is 111.


Example 2:

Input: $n = 6 Output: 0 Since 6 in base 2 is 110, 6 in base 3 is 30 and 6 in base 4 is 12.  Example 3: Input:$n = 8
Output: 1

Since 8 in base 3 is 22.


# The questions

I guess the only question here would be about the limits in the input, like the possibility to have a higher bound or so.

# The solution

I started with a pure brute force implementation, only to figure that even numbers since 8 on are all Brazilian, only to find out that this can be generalized… so I ended up with a hybrid solution in Perl:

#!/usr/bin/env perl
use v5.24;
use warnings;
use experimental 'signatures';
no warnings 'experimental::signatures';

my @candidates = @ARGV ? @ARGV : (7, 6, 8);
for my $candidate (@candidates) { my$bb = is_brazilian($candidate); say "$candidate -> ", ($bb ? 1 : 0), " #$bb";
}

sub is_brazilian ($n) { for my$p (2, 3, 5, 7, 11, 13, 17, 19) {
next if $n %$p;
my $b =$n / $p - 1; next if$b <= $p; return$b;
}
return is_brazilian_bf($n); } sub is_brazilian_bf ($n) {
BASE:
for my $b (reverse 2 ..$n - 2) {
return $b if is_brazilian_with($n, $b); } return 0; } sub is_brazilian_with ($n, $b) { use integer; my$digit = $n %$b;
while ($n > 0) { return 0 if$digit != $n %$b;
$n /=$b;
}
return 1;
}


The reasoning goes along these lines: if the number $n$ is divisible by some prime $p$, we can express it like this:

$n = k \cdot p = (k - 1) \cdot p + p = b \cdot p + p$

Now, if $p$ is strictly lower than $k - 1$, this is exactly how we would express $n$ in base $b = k - 1$, i.e. $n$ is brazilian because it can be expressed as $pp$ in base $k - 1$. As an example, let’s take integer 35:

$35 = 7 \cdot 5 = 6 \cdot 5 + 5$

i.e. it can be expressed as $55$ in base 6.

We MUST check that $p < b = k - 1$, because of how base-$b$ numbers work, i.e. the “digits” can only be in the integer range between 0 and $b - 1$. Hence, for each prime $p$ this mechanism will work only for bases that are greater than $p$, i.e. $b \geq p + 1$. This allows us finding out the minimum value for which this trick applies:

$n_{min} = b \cdot p + p = (p + 1) p + p = (p + 2) p \approx p^2$

The last approximation is increasingly true as $p$ increases, because the term 2 gets proportionally less significant. This tells us that for any number $n$, it only makes sense to check this trick with primes up to about $\sqrt{n}$, which is anyway what we would do anyway for testing divisibility.

In the Perl code, though, we try a few initial primes; if we have success (which happens a lot) we exit early, otherwise… we go brute force with is_brazialian_bf. To be honest, I need some more requirements to get ntheory into the lot and have a more proper source of primes to use.

This was easy to transfom and generalize in Raku, thanks to the is-prime built-in. The brute force part here is baked directly into the is-brazilian that takes one single parameter.

#!/usr/bin/env raku
use v6;
sub MAIN (*@args) {
my @candidates = @args ?? @args !! (7, 6, 8);
s{\,} = '' for @candidates;
for @candidates -> $candidate { my$bb = is-brazilian($candidate); "$candidate -> {$bb ?? 1 !! 0} #$bb".put;
}
}

multi sub is-brazilian (Int() $n where * > 3) { for 2 ..$n.sqrt -> $p { next unless$p.is-prime;
next if $n %$p;
my $d = ($n / $p - 1).Int; next if$d <= $p; return$d;
}
for (2 .. $n - 2).reverse ->$b {
return $b if is-brazilian($n, $b); } return 0; } multi sub is-brazilian (Int:D$n is copy where * > 3, Int:D $b where * > 1) { my$digit = $n %$b;
while $n > 0 { return 0 if$digit != $n %$b;
$n = (($n - $digit) /$b).Int;
}
return 1;
}


I hope this is enough, because at this point I can only recommend that you stay safe!