The Pisano period, denoted π(n), is the period during which the Fibonacci sequence repeats after reducing the original sequence modulo n. More generally, one can similarly define Pisano periods for any linear recurrence sequence; in this paper we compare the Pisano periods of certain linear recurrence sequences with the Pisano periods of the Fibonacci sequence. We first construct recurrence sequences, defining the initial values as integers from 2 to 1000 and second values as 1. This paper discusses how the constructed sequences are related to the matrix M = [(first row) 1 1 (second row) 1 0] reduced modulo n. We offer a proof to show that the order of M is equal to the Pisano period of the Fibonacci sequence reduced modulo n. Further, we provide data showing that there are few discrepancies between the order of M and the Pisano periods of the constructed sequences reduced modulo n, for n from 2 to 1000. Finally, we detail progress made in the analysis of the comparison between the Pisano periods of the Fibonacci and Lucas sequences.
Willrich, Katherine, "Pisano Periods: A Comparison Study" (2019). Mathematics Senior Capstone Papers. 10.