Mathematicians Pursue Numbers That Might Uncover the Boundaries of Mathematics

Amateur mathematicians find themselves ensnared in a vast numerical puzzle.

This conundrum stems from a deceptively simple query: How can one determine if a computer program will execute indefinitely? The roots of this question trace back to mathematician Alan Turing, who in the 1930s demonstrated that computer algorithms could be represented through a hypothetical “Turing machine” that interprets and records 0s and 1s on infinitely long tapes, utilizing more intricate algorithms that necessitate additional states and adhering to a specific set of instructions.

<p>For numerous states, like 5 or 100, the corresponding Turing machines are finite; however, it remains uncertain how long these machines will operate. The longest conceivable run time for each state count is termed the busy beaver number or BB(n), and this sequence grows exceedingly rapidly. For instance, BB(1) equals 1, while BB(2) is 6, and the fifth busy beaver number reaches 47,176,870.</p>
<p>The exact value of the next busy beaver number, the sixth, has not yet been determined, but the online community known as the Busy Beaver Challenge is <a href="https://bbchallenge.org/story">on the verge of discovery</a>. They succeeded in uncovering BB(5) in 2024, concluding a 40-year search, currently attributed to a participant called "MXDYS." <a href="https://bbchallenge.org/1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE">It must be at least as vast as a significantly large value, making even its explanation a challenge.</a></p>
<p>"This number surpasses the realm of physical comprehension. It's simply not intriguing," states <a href="https://www.sligocki.com/about/">Shawn Ligokki</a>, a software engineer and contributor to the Busy Beaver Challenge, who likens the search for Turing machines to fishing in uncharted mathematical oceans filled with strange and elusive entities lurking in the darkness.</p>
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<p>The threshold for BB(6) is so immense that it necessitates a mathematical framework that goes beyond exponents, demanding the raising of one number to another x power, or n<sup>x</sup>2 days etc. For instance, 2*2*2 equals 8. The concept of a tetrol sometimes represented as <sup>x</sup>n <sup>3</sup>2 is raised to the second power and subsequently elevated to the second power again, resulting in a value of 16.</p>
<p>Surprisingly, MXDYS posits that BB(6) is at least two tetroized. The number 2 is illuminated by multiplying two tetroized, resulting in nine. In comparison, the estimated quantity of all particles in the universe seems diminutive, according to Ligokki.</p>

<p>However, the significance of the busy beaver numbers extends beyond their sheer size. Turing established that certain Turing machines must exist that cannot reliably predict behavior under the ZFC theory. This notion was influenced by the mathematician Kurt Gödel's "Incompleteness Theorem," which concluded that using the ZFC rules, it is impossible to affirm that the theory is entirely devoid of contradictions.</p>
<p>"The exploration of busy beaver numbers provides a concrete, quantitative representation of a phenomenon identified by Gödel and Turing almost a century ago," remarks <a href="https://www.cs.utexas.edu/people/faculty-researchers/scott-aaronson">Scott Aaronson</a> from the University of Texas at Austin. "I’m not merely suggesting that a Turing machine could displace ZFC capabilities and ascertain its behavior after a finite stage; rather, is this already occurring with machines possessing six states, or is it restricted to machines with 600 states?" Research has confirmed that BB(643) does eliminate ZFC theory, though numerous examples remain to be investigated.</p>
<p>"The busy beaver problem offers a comprehensive scale to navigate the forefront of mathematical understanding," states Tristan Stérin, a computer scientist who initiated the Busy Beaver Challenge in 2022.</p>
<p>In 2020, <a href="https://scottaaronson.blog/?p=4916">Aaronson wrote</a> that the busy beaver feature "encapsulates most intriguing mathematical truths within its first 100 values," and BB(6) is no exception. It seems to relate to Korizat's hypothesis, an esteemed unsolved mathematical problem that conducts simple arithmetic operations with numbers to determine if they resolve to 1. The discovery of a machine that halts might imply that the particular version of the hypothesis possesses a computational proof.</p>

<p>The numerical challenges that researchers encounter are astonishing in scale, yet the busy beaver framework serves as a tangible measurement tool that otherwise becomes a nebulous expanse of mathematics. In Stérin’s perspective, this aspect continues to captivate many contributors. He estimates that numerous individuals are presently dedicated to the discovery of BB(6).</p>
<p>Thousands of "hold-out" Turing machines remain unexamined for halting behavior, he notes. "There might exist a machine unbeknownst to you lurking just around the corner," Ligokki asserts. In essence, it exists independently of ZFC and lies beyond the boundaries of contemporary mathematics.</p>
<p>Is the precise value of BB(6) also lurking nearby? Ligokki and Stérin acknowledge their reluctance to forecast the future of busy beavers, yet recent achievements in defining boundaries give Ligokki a sense of "intuition that it’s approaching closer."</p>

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