One theoretical motivation for its existence is that it has the fundamental property that we are immediately certain it will terminate, unlike while loops with arbitrary conditions.
Primitive recursive functions are the complexity class that divides those two.
There is no fundamental difference between them, a quantum algorithm is a quantum circuit, which can be seen as a super complicated quantum gate.
is the largest number of 1's written by a halting -state Turing machine on a tape initially filled with 0's.
The Boundary of Computation by Mutual Information (2023)
Source. The Busy beaver scale allows us to gauge the difficulty of proving certain (yet unproven!) mathematical conjectures!
To to this, people have reduced certain mathematical problems to deciding the halting problem of a specific Turing machine.
A good example is perhaps the Goldbach's conjecture. We just make a Turing machine that successively checks for each even number of it is a sum of two primes by naively looping down and trying every possible pair. Let the machine halt if the check fails. So this machine halts iff the Goldbach's conjecture is false! See also Conjecture reduction to a halting problem.
Therefore, if we were able to compute , we would be able to prove those conjectures automatically, by letting the machine run up to , and if it hadn't halted by then, we would know that it would never halt.
Of course, in practice, is generally uncomputable, so we will never know it. And furthermore, even if it were computable, it would take a lot longer than the age of the universe to compute any of it, so it would be useless.
However, philosophically speaking at least, the number of states of the equivalent Turing machine gives us a philosophical idea of the complexity of the problem.
The busy beaver scale is likely mostly useless, since we are able to prove that many non-trivial Turing machines do halt, often by reducing problems to simpler known cases. But still, it is cute.
But maybe, just maybe, reduction to Turing machine form could be useful. E.g. The Busy Beaver Challenge and other attempts to solve BB(5) have come up with large number of automated (usually parametrized up to a certain threshold) Turing machine decider programs that automatically determine if certain (often large numbers of) Turing machines run forever.
So it it not impossible that after some reduction to a standard Turing machine form, some conjecture just gets automatically brute-forced by one of the deciders, this is a path to
CIA 2010 covert communication websites USA spying on its own allies by 
Ciro Santilli 37 Updated 2025-08-08
Ciro Santilli 37 Updated 2025-08-08Being Brazilian, Ciro Santilli was particularly curious about the existence of a Brazil-focused mentioned in the Reuters article, as well as in other democracies.
WTF the CIA was doing in Brazil in the early 2010s! Wasn't helping to install the Military dictatorship in Brazil enough!
Here are the websites likely targeting democracies based on their language and content found so far, defining a democracy as a country with score 7.0 or more in the Democracy index 2010:In English, so more deniable:"Almost democracies":Ciro couldn't help but feel as if looking through the Eyes of Sauron himself!
- France (6: affairesdumonde.com, guide-daventure.com, lesummumdelafinance.com, football-de-luxe.com, romulusactualites.com, suparakuvi.com)
- Germany(2: dedrickonline.com, neighbour-news.com)
- Italy (2: attivitaestremi.com, garanziadellasicurezza.com, podisticamondiale.com)
- Spain (3: armashoy.com, montanismoaventura.com, ordenpolicial.com)
- Brazil (2: noticiasmusica.net, vejaaeuropa.com)
- South Korea (1: economicnewsbuzz.com)
- Poland (1: boxingstop.net)
- Japan (1: snapnewsfront.net)
- Canada (2: kanata-news.com, mynewscheck.com)
- Philippines (1: half-court.net)
- India (1: amishkanews.com)
- Croatia (2: european-footballer.com, stara-turistick.com)
- Thailand (1: thefairwaysaregreen.com)
- Peru (1: todosperuahora.com)
It is worth noting that democracies represent just a small minority of the websites found. The Middle East, and Spanish language sites (presumably for Venezuela + war on drugs countries?) were the huge majority. But Americans have to understand that democracies have to work together and build mutual trust, and not spy on one another. Even some of the enlightened people from Hacker News seem to not grasp this point. The USA cannot single handedly maintain world order as it once could. Collaboration based on trust is the only way.
Snowden's 2013 revelations particularly shocked USA "allies" with the fact that they were being spied upon, and as of the 2020's, everybody knows this and has "stopped caring", and or moved to end-to-end encryption by default. This is beautifully illustrated in the 2016 film "Snowden" when Snowden talks about his time in Japan working for Dell as an undercover NSA operative:
NSA wanted to impress the Japanese. Show them our reach. They loved the live video from drones. This is Pakistan right now [video shows American agents demonstrating drone footage to Japanese officials]. They were not as excited about that we wanted their help to spy on the Japanese population. They said it was against their laws.And we did not stop there. Once we owned their communications systems, we started going after the physical infrastructure.We'd slip these little sleeper programs into power grids, dams, hospitals. The idea was that if the day came when Japan was no longer an ally, it would be "lights out".But Austria?!
Another noteworthy scene from that movie is Video 2. "Aptitude test on communication networks scene from the 2016 Snowden film", where a bunch of new CIA recruits are told that:thus somewhat mirroring what actually happened with these real world websites.
Each of you is going to build a covert communications network in your home city [i.e. their fictitious foreign target location written on each person's desk such as Berlin, Istanbul and Bangkok, not necessarily where they were actually born], you're going to deploy it, backup your site, destroy it, and restore it again.
The prototypical example is the Busy beaver function, which is the easiest example to reach from the halting problem.
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