Amateurs are approaching the colossal number that challenges the limit of modern mathematics.
A group of amateur mathematicians will be approaching the discovery of such a large number that it will challenge the limits of mathematical knowledge-a milestone that can reveal to where, effectively, the ability of contemporary mathematics to understand the infinite.
The riddle was born from a not so complicated question: “How to know if a computer program will run forever?”
The answer dates back to the 1930s, when the mathematician demonstrated that any algorithm can be represented by a “turing machine” – a theoretical device that reads and writes zeros and some in an infinite tape, according to a set of instructions, or “states.” The more complex the task, the more states are needed.
From this the function is born Busy Beaver (Bassed Castor), which identifies the largest number of steps than any Turing machine with a specific state number can run before stopping.
This sequence grows explosively, explains: Bb (1) is 1, bb (2) is 6, and bb (5) is already 47.176.870.
The discovery of this last value, made in 2024 – after 40 years behind her – rekindled the enthusiasm around the calculation of BB(6) – Still unknown, but now with new promising clues.
The group, created online in 2022, leads exploration. Recently, the “mxdys” member has established that BB (6) is, at least, as large as a number that can only be described through highly advanced mathematical concepts, such as the tature – A form of iterated exponency that greatly surpasses the limits of traditional notation.
In this case, BB (6) is at least 2 teturated to 2, tetracted to 2, tettable to 9 – a true numerical skyscraper which makes the total number of particles in the universe seems insignificant.
But the relevance of the Busy Beaver function goes beyond the size of the numbers. Inspired by, Turing has shown that there are Turing machines whose behavior cannot be foreseen using ZFC theory axioms – the basic system of modern mathematics. This means that There are intrinsic limits to what can be proven mathematically. BB function makes these limits tangible, allowing mathematicians to ask themselves, for example, if a machine with only six states already challenges ZFC, or if it takes a much larger number of states.
So far, it is known that BB (643) already exceeds ZFC capacity to predict your behavior. But there are many other machines with fewer states yet to study. For computer computer computer, creator of Busy Beaver Challenge, this function offers a complete scale to reflect on the border of mathematical knowledge.
Scott Aaronson, professor at the University of Texas in Austin, tells New Scientist magazine that the first 100 values of the Busy Beaver function may contain a vast amount of “interesting mathematical truths.” BB (6), for example, seems to be linked to the famous – an unresolved problem that involves simple arithmetic operations whose relationship with Turing machines can bring new forms of computational approach to the problem.
