Keith Matthews' 3x+1 page

I spent a considerable amount of time starting in 1981, helped by collaborators Tony Watts, George Leigh and Bob Buttsworth, studying certain aspects of the 3x+1 problem (Collatz problem) and related problems. My starting point was a paper of H. Möller (Über Hasses Verallgemeinerung des Syracuse-Algorithmus (Kakutanis Problem), Acta Arith. 34 (1978) 219-226) which related the 3x+1 problem to 2-adic analysis.

The survey below presents a point of view which makes the 3x+1 problem appear as part of a more general class of problems which are equally tantalizing, but about which one can make accurate predictions.

Starting on 31st October 2005, I made yet another attempt to understand parts of George Leigh's paper A Markov process underlying the generalized Syracuse algorithm, Acta Arithmetica 46 (1986) 125-143, as well as G. Venturini's paper Iterates of number-theoretic functions with periodic rational coefficients (generalization of the 3x+1 problem), Stud. Appl. Math. 86 (1992), no.3, 185-218.

Both authors clearly had great insights into the Markov chain aspects of the generalised 3x+1 mapping. Finally (February 2006) I made some progress and put some of my thoughts online.

A survey of the generalized 3x+1 (Collatz) mapping (pdf 190K).
An html account of the Matthews-Watts work on generalized 3x+1 mappings. This also gives some proofs omitted from the survey.
An account of some aspects of George Leigh's paper on generalized 3x+1 mappings.
Cycle-finding program for generalized 3x+1 functions.
Some BCMATH examples of generalised 3x+1 mappings.

Email
http://www.numbertheory.org/keith.html

Last modified 16th August 2006