I’ve been doing the Weekly
Challenges. The
latest
involved permutations and primality testing. (Note that this is
open until 6 March 2022.)

Task 1: Missing Permutation

You are given possible permutations of the string 'PERL'.

Write a script to find any permutations missing from the list.

My basic approach is to make a list of all possible permutations (of
the first argument in the list, rather than making it specific to one
string), then delete from it those in the input list.

A fairly sharp dividing line separates the languages with some sort of
built-in permutation engine (Python, Ruby, Raku) or readily-available
module (Perl, Rust) and those without (the latter being PostScript,
JavaScript, Kotlin and Lua). For those last I wrote a non-recursive
implementation of Heap's
Algorithm (dodging
about a bit for Lua with its 1-based pseudo-arrays). For the others it
was rather easier. The cleanest code is probably Python with
`itertools`

:

```
def missingpermutations(lst):
perms=set("".join(x) for x in permutations(lst[0]))
for x in lst:
perms.discard(x)
return list(perms)
```

But just for fun here's a permutor in PostScript:

```
/permute { % [array] {proc} permute runs proc on each permutation of array
7 dict begin
/subproc exch def
/a exch def
/n a length def
/c [ n { 0 } repeat ] def
mark a subproc cleartomark
/i 0 def
{
i n ge {
exit
} if
c i get i lt {
i 2 mod 0 eq {
0 i permute.swap
} {
c i get i permute.swap
} ifelse
mark a subproc cleartomark
c i get 1 add c exch i exch put
/i 0 def
} {
c i 0 put
/i i 1 add def
} ifelse
} loop
end
} bind def
/permute.swap {
/bi exch def
/ai exch def
a ai get
a bi get
a exch ai exch put
a exch bi exch put
} bind def
```

Task 2: Padovan Prime

A Padovan Prime is a Padovan Number that's also prime.

In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values.

P(0) = P(1) = P(2) = 1

and then followed by

P(n) = P(n-2) + P(n-3)

Write a script to compute first 10 distinct Padovan Primes.

Opinions differ; Wikipedia quotes this definition, while the
OEIS starts its system with `(1, 0, 0)`

.
But 0 and 1 are canonically non-prime anyway.

With Raku I was able to use its lazy sequence generator:

```
sub padovanprime($ct) {
my $pp=SetHash.new;
for (1, 1, 1, -> $a, $b, $c { $a + $b } ... *) -> $padovan {
if (isprime($padovan)) {
$pp{$padovan}=True;
if $pp.elems >= $ct {
last;
}
}
}
return [$pp.keys.sort];
}
```

But, sadly, not to rely on the built-in and quite fast `is-prime`

function. This will reliably return false if the number is known *not*
to be prime, but returns true for either a prime or a number with
unknown primality. (Perl's `Math::Prime::Util`

at least has a
three-state return for (no, unknown, yes).) With numbers of this size
it probably won't matter, but as a result I imported the
primality-testing code I'd written for other languages.

(Normally I'd use a sieve of Eratosthenes as in previous Challenges,
but since I don't know in advance how large the numbers will be I fall
back on lightly-optimised trial division.)

```
sub isprime($candidate) {
```

Check the easy cases first.

```
if (!is-prime($candidate)) {
return False;
} elsif ($candidate==2) {
return True;
} elsif ($candidate==3) {
return True;
} elsif ($candidate % 2 == 0) {
return False;
} elsif ($candidate % 3 == 0) {
return False;
}
```

Failing that, start checking divisors of the form `6n±1`

until we
either exceed the square root of the candidate or find an even divisor.

```
my $anchor=0;
my $limit=floor(sqrt($candidate));
while (True) {
$anchor+=6;
for ($anchor-1,$anchor+1) -> $t {
if ($t > $limit) {
return True;
}
if ($candidate % $t == 0) {
return False;
}
}
}
}
```

Although this is basically the same code I'd written for other
languages, it was desperately slow in Raku, reliably taking about 58s
to complete the test cases on my unloaded reference machine. Optimised
compiled Rust took 0.2s; Kotlin took 0.45s even with Java runtime
startup; Javascript under Node, 0.8s; Ruby, 1.4s; Rust unoptimised
including compilation time 1.5s; Lua, 1.9s; Perl, 3.2s; Python, 6.7s,
PostScript, 7.0s.

(Yeah, one might start to think that Rust is actually not a terrible
scripting language…)

Full code on
github.

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