Hittills har man från politiskt håll kommit med få förslag som berör annat än rent organisatoriska åtgärder för skolan (mindre klasser, när betyg ska ges, läxor eller inte). Enligt vår mening är det viktigare att diskutera hur betygen sätts och hur läxorna ges skriver Marcus Nilsson, prefekt för Institutionen för matematik och Torsten Lindström, professor i matematik.

We consider the dynamics of x ↦ xⁿ, where n ≥ 2 is an integer, over the multiplicative group modulo p^k, where k is a positive integer and p an odd prime. This paper is a review of earlier results by the author, but new results are also contained. Possible applications to pseudorandom number generation will be discussed. The main results are a description of the preperiodic points and an algorithm to find the longest possible cycle. The preperiodic points form trees, all isomorphic as graphs to the preperiodic points of the fixed point 1. When n is a prime, different from p, we can describe the tree structure completely. A formula for the length of the longest cycle is presented. We can find one of the longest cycles of the monomial system using a primitive root modulo p^k as an initial value.

We first study monomial systems on the n-torus modulo a prime number p.  We investigate the dynamics by using group structures and directed graphs.  Under certain conditions the systems we get by lifting to the p-adic  n-torus inherits much of the structure of the system modulo p.  We also  discuss the characters of the periodic points.

The aim of this paper is to provide some results that

are useful when monomial dynamical systems are used for pseudorandom number generation. We prove that if we choose the seed to be a primitive root modulo $p^k$ then we get

the longest possible period of the random sequence modulo

$p^k$. An explicit expression for the length of the longest period is

also provided.

We investigate perturbed monomial dynamical system over F_p given by iterations of x↦xⁿ+c mod p, where c ∈ F_p. Instead of study the systems one at a time we study all of them at the same time. The complex distibution of periodic points is visualized in the so called Periodic Point Diagram, which can be seen as a discrete version of the classical Bifurcation Diagram. We also prove some general results about the distribution of periodic points. We end the article with a conjecture about the total number of periodic points.