In this work we are going to investigate a factorization method that was invented by John Pollard. It makes possible to factorize medium large integers into a product of prime numbers. We will run a C++ program and test how do different parameters affect the results. There will be a connection drawn between the Pollard's rho method, the Birthday paradox and the Floyd's cycle finding algorithm. In results we will find a polynomial function that has the best effectiveness and performance for Pollard's rho method.

This paper is a bachelor thesis in Applied Mathematics at Linnaeus University. The goal of this thesis is to find a structure in sets of discrete quadratic dynamical systems modulo a number, with a main focus on 1-, and 2-periodic points in sets of systems modulo an odd prime. The amount of 1-, and 2-periodic points in such sets is numerically investigated and is proven directly to be p, respectively p-1. Furthermore, the sets are visualized using a diagram, where some apparent structures are noticed, and later explained. Finally, through numerical investigations, an expression for the amount of 2-periodic points in a system modulo a composite number is also perceived, and expressed as a conjecture. Proving the conjecture should be done in a future project.