Mathematicians are chasing a number that may reveal the edge of maths

Amateur mathematicians are on the brink of unraveling an enormously vast number – one that challenges the limits of what is comprehensible within the realm of contemporary mathematics.

The journey begins with a deceptively simple query: how can one determine if a computer program will continue to run indefinitely? The exploration of this issue traces back to the work of mathematician Alan Turing. In the 1930s, Turing demonstrated that any computer algorithm could be replicated by envisioning a basic “Turing machine” that manipulates 0s and 1s on an infinitely long tape based on a series of instructions known as states, with more intricate algorithms necessitating a greater number of states.

For each quantity of states, such as 5 or 100, there exists a finite number of corresponding Turing machines, yet the duration for which each of these machines must operate remains uncertain. The maximum possible runtime for each state number is denoted as the Busy Beaver number or BB(n), and this sequence expands at an astonishing rate: BB(1) is 1, BB(2) is 6, while the fifth Busy Beaver number is a staggering 47,176,870.

The specific value of the succeeding Busy Beaver number, the sixth one, remains elusive, but a dedicated online community known as the Busy Beaver Challenge is actively engaged in uncovering it – having successfully revealed BB(5) in 2024, culminating a 40-year quest. Presently, a member named “mxdys” has ascertained that it must be at least as colossal as a number of such immense proportions that elucidating it necessitates a detailed explanation.

“This number transcends physical reality by leaps and bounds,” remarks Shawn Ligocki, a software engineer and contributor to the Busy Beaver Challenge. He likens the exploration of all potential Turing machines to casting a line into a profound mathematical ocean teeming with peculiar and enigmatic snippets of code.

The novel boundary for BB(6) is so vast that it demands a mathematical vernacular surpassing exponentiation – the act of raising one number n to the power of another x, represented as n^x, such as 2³, which equals 2*2*2 = 8. Initially, there is tetration, sometimes notated as ^xn, which entails repeated exponentiation, thereby ³2 would signify 2 raised to the power of 2, raised to the power of 2, yielding 16.

Remarkably, mxdys has demonstrated that BB(6) is at minimum 2 tetrated to the 2 tetrated to the 2 tetrated to the 9, an intricate tower of iterated tetration, with each tetration constituting a stack of iterated exponentiation. In comparison, the total number of particles in the universe appears minuscule, according to Ligocki.

However, the significance of the Busy Beaver numbers transcends their astronomical size. Turing established that there must exist certain Turing machines whose behaviors cannot be anticipated within the confines of ZFC theory, a foundational framework supporting conventional modern mathematics. His inspiration stemmed from mathematician Kurt Gödel’s “incompleteness theorem”, which illustrated that the principles of ZFC itself cannot definitively prove the theory to be completely devoid of contradictions.

“The investigation of Busy Beaver numbers is rendering the phenomena unveiled by Gödel and Turing nearly a century ago quantifiable and tangible,” states Scott Aaronson at the University of Texas at Austin. “Instead of merely positing that Turing machines must surpass the capacity of ZFC to ascertain their behavior beyond a certain point, we can now inquire whether this occurs with 6-state machines or solely with 600-state machines?” Researchers have substantiated that BB(643) would outstrip ZFC theory, although many of the smaller numbers remain unexplored.

“The Busy Beaver problem furnishes a comprehensive gauge for contemplating the frontier of mathematical knowledge,” affirms computer scientist Tristan Stérin, who inaugurated the Busy Beaver Challenge in 2022.

In 2020, Aaronson asserted that the Busy Beaver function “likely encapsulates a substantial portion of all captivating mathematical truths within its initial hundred values,” and BB(6) is no exception. It appears to be intertwined with the Collatz conjecture, an emblematic unsolved mathematical enigma revolving around repetitive arithmetic operations on numbers and ascertaining whether they ultimately converge to 1. The quest for BB(6) appears to be linked to a Turing machine that must emulate certain steps of this conundrum to terminate. A machine that halts would signify the existence of a computational proof for a variant of the conjecture.

The numbers under scrutiny are staggering in their enormity, yet the Busy Beaver framework furnishes a measuring rod for what would otherwise be an inscrutable realm of mathematics. Stérin contends that this is what retains the interest of numerous contributors, even though most are not academicians. He estimates that there are presently a few dozen individuals consistently striving to unveil BB(6).

There remain several thousand Turing machines that have not undergone scrutiny for their halting behavior, he notes. “Just around the corner, there could lurk a machine that is unknowable,” Ligocki remarks, alluding to a machine independent of ZFC and transcending the boundaries of contemporary mathematics.

Could the precise value of BB(6) also be on the cusp of revelation? Ligocki and Stérin both maintain that they refrain from forecasting Busy Beaver’s future, yet the recent success in bounding the number instills in Ligocki an “inkling that there are further revelations on the horizon,” he asserts.