PWC155 - Pisano Period


On with TASK #2 from The Weekly Challenge #155. Enjoy!

The challenge

Write a script to find the period of the 3rd Pisano Period.

In number theory, the nth Pisano period, written as π(n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats.

The Fibonacci numbers are the numbers in the integer sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, ...

For any integer n, the sequence of Fibonacci numbers F(i) taken modulo n is periodic. The Pisano period, denoted π(n), is the value of the period of this sequence. For example, the sequence of Fibonacci numbers modulo 3 begins:

0, 1, 1, 2, 0, 2, 2, 1,
0, 1, 1, 2, 0, 2, 2, 1,
0, 1, 1, 2, 0, 2, 2, 1, ...

This sequence has period 8, so π(3) = 8.

The questions

Challenges lately are quite tight in their requirements that questions are hard to think. I mean, if it were a more general question asking for the $n$th number, I could have asked what the maximum value for $n$ would be…

I know it’s 3, I’ll assume it’s not that big 😄

The solution

Calculating the period by taking the remainder of the Fibonacci sequence is the same as calculating each new item in the sequence directly from the previously calculated reminders. In other terms, if we define:

\[x_n := \{v: v = x + k \cdot n, k \in \mathbb{Z}\}\]

that is the usual definition of remainder class modulo $n$, then we have:

\[(x + y)_n = x_n + y_n\]

In considering $x_n$, we can take any of the values $v$ in the class, although usually the lowest non-negative value is used for simplicity.

In other terms, we can calculate the modulo operation as soon as we sum two items, and forget about calculating the original Fibonacci sequence.

This means that sequences MUST be periodic, because any time we end up with the same pair of consecutive values, the sequence is going to repeat. With only $n$ possible remainders, there are only $n^2$ possible pairs of consecutive values, so at a certain point we MUST hit a pair that we already saw.

Additional demonstrations show that it’s OK to take the initial values in the sequence as the ones to look for periodicity, so we will concentrate on looking for $0_n$ followed by $1_n$. This is important because the periodic part might be preceded by some a-periodic head, but this is not the case here.

Let’s look at the Perl code:

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

say pisano_period(shift // 3);

sub pisano_period ($n) {
   my ($fl, $fh) = (0, 1 % $n);
   my $pp = 0;
   while ('necessary') {
      ($fl, $fh) = ($fh, ($fl + $fh) % $n);
      return $pp if $fl == 0 && $fh == 1 % $n;

The astute reader will have noted that we’re doing 1 % $n while initializing the two Fibonacci state variables. This is because $n = 1$ is a valid input, and in that case the remainder is $0$, not $1$.

We might just as well written this:

sub pisano_period ($n) {
    return 1 if $n == 1;
    my ($fl, $fh) = (0, 1);

Which leads us to Raku, where multi sub are a way to deal with special cases:

#!/usr/bin/env raku
use v6;

sub MAIN (Int:D $n = 3) { put pisano-period($n) }

multi sub pisano-period (1)  { return 1 }
multi sub pisano-period ($n) {
   my ($fl, $fh) = 0, 1;
   my $pp = 0;
   loop {
      ($fl, $fh) = $fh, ($fl + $fh) % $n;
      return $pp if $fl == 0 && $fh == 1;

Stay safe folks!

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