Minna Litzén: Fibonacci modulo m

Abstract

A periodic sequence arises when the elements of the Fibonacci sequence are taken modulo m. This thesis is about the period of the Fibonacci sequence modulo m, where we pay attention to its properties. Several theorems are shown with methods from number theory and algebra. We also address a conjecture that has been unproven for a longer amount of time. The problem of calculating the length of the period can be linked to the discrete logarithm problem, which has led to studies on the period having several possible areas of application. We will get familiar with the Legendre symbol and the law of quadratic reciprocity to calculate upper bounds on the period, we will also touch on the concept of a splitting field. At last, the properties of the period of the Fibonacci sequence are generalized to the period of a general second-order linear recurrence sequence.