I had a quick read. The paper states in the conclusion "The more we increase the number of iterations k, the more proportion decreases, but the number of integers starting the different trajectories keeps increasing and tends towards infinity as k tends towards infinity."
This statement is not proven, and is not supported by the previous theorems. Thus the following statement "We therefore conclude that the 3x+ 1 conjecture is false" remains an unproven assertion.
Essentially the paper argues that the density F(k) of trajectories at least as long as 'k' gets ever smaller as k increases, without ever reaching zero, and thus there exists at least one value that has an infinity long trajectory.
But that's as false assertion. Just because F(k) is non-zero for every finite value of k does NOT imply that the lim as k tends to infinity of F(k) is non-zero, and consequently does not prove the existence of an infinity long trajectory.
TL;DR: paper applies intuition on finite sequences to infinite sequences and gets it wrong.
Me, 10$K/month. Because of AdSense revenue, I took many hours building a new website (online since last month) and hope that website it will eventually make some money. I really don't like having ads on website I visit but don't use AdBlocker (hide it behind other windows). I think WikWik.org is a great site (I'm the author) and for sure would not have exist without AdSense opportunity.
No, didn't think to do that. Part of the reason I went to all this trouble, though, was to go through the process of building something myself, that I could run myself--and that, by extension, anyone else could do, too.
Hello from Quebec,
I am on Hacker News, as a big reader not commenting. My online business, is profitable, it make all my income, an ok salary for me :-). I have read the book '4 hours week' and work only few hours a week. The business start with a shareware game (1990), quit my day job (2002) to create more sharewares, fail at the first one (the password/unlock was hack the first week). So I come with the idea of having a client/server game (2004) (harder to hack). That work enough to make a small salary. Then I build another client/server game (2011), almost the same as the first one but localized in 3 languages. Then I received a lawyer letter (2011) to close both of my online sites. I did make some modifications, after 2 years they leaves me alone... Being afraid of closing, I was looking for a plan B (2012), I works hard on web sites that have a lot of visitors to make money with AdWords, it works. Now half of the revenue come from the 2 online games, and the other half come from AdWords. The shareware, online games and web sites are all related to a very popular crossword game.
The 3x+1 Conjecture is false Text in English : https://arxiv.org/abs/2104.10681