An Engineering and Statistical Look at the Collatz (3n + 1) Conjecture

Abstract

The famous (3n + 1) or Collatz conjecture has admitted some progress over the last several decades towards the conclusion that the conjecture is true (i.e. that all Collatz sequences will eventually reach a value of one), but has stubbornly resisted a proof. Many professional mathematicians have applied an impressive array of machinery to its analysis, and discovered some limits or boundaries to where violations of the Collatz conjecture must be, if they exist. Here we attempt to look at the conjecture differently than most mathematicians or number theorists, i.e. we will study the conjecture from the points of view of a statistician/data scientist and of an engineer. As a statistician or data scientist, we look at Collatz sequences as if they were sequences found in nature, like a collection of time series produced by some natural process, ignoring for the time being their completely deterministic origins. As an engineer, we try to tinker with Collatz sequences to see what makes them tick, and engineer -- tweak and build -- changes to the sequences that likewise do interesting things, after first specifying exactly what we mean by interesting. Although these analyses do not yield a proof of the Collatz conjecture, like other efforts they present reasons to think that the conjecture is probably true, and the hope is that the analyses of sequences similar to Collatz sequences will tell us something about the nature of Collatz conjecture itself, and/or lead us into interesting and fruitful new directions. Indeed, many of the phenomenon revealed in the sequences of these Collatz-like programs are arguably more interesting than, and suggest conjectures that are as interesting as, the Collatz conjecture itself.