Analyzing the Pisano Period

Algebra 2 Research Paper

Analyzing the Pisano Period

The Fibonacci sequence is known for its seemingly random sequence of numbers that go on forever. Through very simple processes, you get a set of numbers that goes on forever. However, after closer examination, you see many of its special properties that are specific to it and not other sets.

The Fibonacci sequence and the Pisano period were discovered by Leonardo of Pisa, a 13th Century Italian mathematician known for his great works during the Middle Ages (Mastin, 2010). His nickname, was Fibonacci. At the time, Fibonacci was just trying to promote the use of the Arabic numeral system, and tried to show all of its benefits. He wrote a book called ”Liber Abaci” or “Book of Calculations”. In this book, he described something interesting that led to his amazing discovery of the Fibonacci sequence, rabbit breeding. More specifically, the generations of rabbit breeding. Each month, rabbits would mate in a consistent pattern and the growth of population that came from this would increase the number of babies that came each month. His results happened to be 1, 1, 2, 3, 5, 8 and so on. This was the start of the Fibonacci sequence. Now currently, there isn’t very much information about the Pisano period and how exactly he came up with it. However, Fibonacci wasn’t the only one testing out the Fibonacci sequence. In the 1750s, a man by the name of Robert Simson looked at the ratio between each Fibonacci number. What he concluded was that as you go further into the sequence, the ratio approaches the Golden Ratio. The Golden Ratio, also referred to as Phi, is approximately 1 : 1.6180339887.

As I’ve said before, the Fibonacci sequence has many key properties that are specific to it. The Fibonacci sequence is a sequence of numbers in which you find a term by adding the previous two terms of the sequence, with the first two numbers both being one. The classic sequence looks something like this : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, and so on. What intrigued me was that a student by the name Marco Calvo was able to find a very important pattern in the Fibonacci sequence. Every third number was divisible by 2, the third number of the sequence. Every fourth number was divisible by 3, the fourth number of the sequence. Every fifth number was divisible by 5, the fifth number in the sequence (Calvo, 2012, p.150). There is a very noticeable pattern here. Now the reason why this intrigued me is that it opened my eyes to the connections and weird properties there were in the Fibonacci sequence.

Now, because this article has information about division, I was wondering if I can possibly find some type of pattern with the numbers you get by dividing the individual terms of the Fibonacci Sequence. However, just dividing the numbers would just leave me with a set of smaller numbers and fractions that still follow the same rules of the Fibonacci sequence. So instead, I tried modular arithmetic. Now to my surprise, this already existed and it happened to be called the Pisano Period.

The Pisano period is the period in which the sequence of Fibonacci numbers taken modulo n repeats. It is often noted as π(x), with x being the modulo value. For different values of modulo, there are different cycles of number that result from it. Of course, each repeats after a certain amount of numbers. When we refer to the Pisano Period of modulo n, we are talking about the length of the cycle that results from the sequence of Fibonacci numbers taken modulo n. Getting Pisano periods however, are not typically an easy process. There is no equation you can use to find the Pisano period for modulo n. Because of this, the only way we can get the Pisano period, is by taking modulo n of the Fibonacci numbers until you see a sequence. For something like modulo 2, it’s very simple. The cycle is 1, 1, 0, so the Pisano period is 3. However, values such as modulo 511, aren’t quite as simple. In the past few weeks, I’ve tried many different sets of modulo values to make equations with a set of conditions that can make finding the Pisano period, a lot less complicated.

Now currently, we don’t know that much about the period. The most important thing we do know is that the Pisano period is the length of a cycle.Gupta, Rockstroh and Su, were able to find out some very intricate information about the Pisano period. They started out showing the sequence with modulo 11 has a Pisano period of 11 and they also showed the sequence with modulo 7 results in a Pisano period of 16. They also said, “For primes p that are congruent to 1 or 4 (mod 5) the period length of the Fibonacci sequence mod p divides p − 1, while for primes p that are congruent to 2 or 3 (mod 5) the period length divides 2(p + 1),” (Gupta, Rockstroh, Su, 2012, p130). This showed me that there are connections that can be made between the Pisano period length along with the modulo value. However, this also showed me how hard it was going to be for me to set conditions to find equations. The examples given here are very specific and cannot be used for any type of general case.

They also described modulo of 5 and 2 as a special case. They were both numbers that gave a simple period length of 20 and 3.This is extremely special as multiple values of modulo will output the same Pisano period length. However, using a modulo of 5 and 2, you get special cases as they are the only numbers that can get a Pisano period length of their own (Gupta, Rockstroh, Su, 2012, p130). The 2 is pretty easy to figure out as it such a small number that is also often used in coding to differentiate odd from even numbers. Something interesting is that this directly correlates with the research done by Calvo. He said that every third number of the sequence is divisible by 2. Well that’s exactly what’s happening here. Because every third number is divisible by 2, you get a very simple Pisano period of 3. This also happens to be called parity, in which we find out if a number is even or odd based on its remainder, which the Pisano period demonstrates perfectly. Now 5 is a bit more complicated but it also falls under the research of Calbo as the number is still small enough to the point where numbers smaller than it can have similar periods.

Even though many people were able to learn about the Pisano period because of the Fibonacci sequence, we also were able to learn something about the Fibonacci sequence from it. People applied modulo 10 to the Fibonacci sequence, making it so we only have the one's digit to work with. When we got the Pisano period, we found out that it had a length of 60 numbers. So in some way, we can predict at least the one's digit of any Fibonacci number if we’re given the position and of course,