Oneness — The Collatz Conjecture

“Math­e­mat­ics is not yet ready for such problems.”

–Paul Erdős, a Hun­gar­ian math­e­mati­cian  known for his eccen­tric­ity (pun not intended! ), and over­all awe­some­ness, express­ing his opin­ion about the Col­latz Con­jec­ture in 1985, and offer­ing a prize of $500, from his own pocket, for the same.

The hottest news in Math town is the recently sub­mit­ted proof for the Col­latz Con­jec­ture. But do you know what it is?

Take any nat­ural num­ber n. If n is even, divide it by 2 to get n / 2, if n is odd mul­ti­ply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called “Half Or Triple Plus One”, or HOTPO) indef­i­nitely. The con­jec­ture is that no mat­ter what num­ber you start with, you will always even­tu­ally reach 1. The prop­erty has also been called one­ness.

Source: Wikipedia

The sequence of num­bers obtained using HOTPO are known as “Hail­stone sequences”. {The con­ver­gence of many such sequences, is depicted in the xkcd comic too! }

This decep­tively sim­ple sound­ing con­jec­ture — which remained unsolved for 74 years — may have been was attempted to be proven by Ger­hard Opfer who was a stu­dent of Collatz.

You can read the proof here or there.

(As of June 5) Error[s] have been found in Opfer’s proof. User “atara_x_ia” says:

The proof depends on the con­struc­tion of an anni­hi­la­tion graph. It is proven that this graph is infi­nite, but in order to com­plete the proof it must be proven that the graph con­tains all of the pos­i­tive integers.

More details, and dis­cus­sions on the var­i­ous inter­est­ing fea­tures of the Col­latz Con­jec­ture, are yet to be cov­ered here.

Statu­tory Note: A lit­tle jar­gon in what fol­lows, along with a pretty picture…

If we map the Col­latz conjecture’s oper­a­tion into a sin­gle func­tion, we can iter­ate that for val­ues on the com­plex plane, and obtain a frac­tal (where the black regions include num­bers whose orbit on the repeat­ing the func­tion, is bounded) — and frac­tals being the same thing we men­tioned often involve a lot of Phi! For now, I will leave you to appre­ci­ate the inter­est­ing­ness of the frac­tal, but we will cer­tainly come back to it soon! If you want to under­stand it bet­ter, you could try out your hand (or head, actu­ally) at Wikipedia, or at an analy­sis by Nathaniel John­ston.
A more com­pre­hen­sive expla­na­tion on how this pic­ture hap­pens to become, is also yet to follow!