A Polynomial Analogue of the 3n + 1 Problem

Abstract

viewed as mapping the integers to itself. The famous 3n + I conjecture postulates that after a finite number of iterations one always arrives at the value 1. This has been shown to be true for all n < 240. Whether it converges to 1 for every positive integer n is an open problem, see Lagarias [3]. The intricate relationship between the ring Z of integers and the polynomial ring Fq[x] of polynomials in a single variable x over the finite field Fq has been studied extensively. This discussion has included many topics such as comparisons between prime integers and irreducible polynomials, Goldbach type problems in both the in teger and polynomial settings (Effinger and Hayes [1]), and the study of twin primes and twin irreducibles and their densities, see [2]. In [2], the authors allude to the fact that in many (but not all) cases an analogous problem in the polynomial setting may be easier to resolve than the original problem. The result presented here indeed provides an example of that phenomenon. We start with the binary field F2 since this is the case most analogous to the integer version, see [2]. In the integer case, the first two primes are of course 2 and 3. In the F2[x] setting, the first two irreducible polynomials are x and x + 1. The analogue to the 3n + 1 problem for polynomials is the map Ci(f(x)) (x + l)f(x) + l if /(0)^0 fix) JK } if/(0) = 0 x