One reason BB(6) may never be pinned down: a tiny Turing machine called Antihydra. The Busy Beaver hunters proved BB(5) = 47,176,870 in 2024. For six states the record keeps climbing (mxdys's June 2025 champion runs for more than a tower of 2s), but the real obstacle is a handful of "cryptids": machines whose behavior people understand perfectly well and still can't decide. Antihydra, found by mxdys in June 2024, reduces to a two-line game. Start at 8 and keep replacing n with floor(3n/2). Keep a score: +2 every time you land on an even number, -1 on an odd one. The machine halts if the score ever drops below zero. If the parities behaved like coin flips, the score would drift upward by half a point per step, and the chance of ever going negative is tiny. So almost everyone expects it to run forever. Proving that, though, means proving the parities of the iterates of 3n/2 aren't secretly biased. That sits right next to Mahler's 3/2 problem from 1968 (is there any number x where the fractional part of x(3/2)^n stays below 1/2 forever?), which is still open, and it has the same flavor as Collatz. What I like about this: "what's the most work a 6-state machine can do" sounds like a computer-science question with a finite answer. It turns out to contain an old number theory problem where, after nearly 60 years, the best results are still partial. My bet is that BB(6) gets a proof only if someone settles a question about powers of 3/2 first, not the other way around. Sources: bbchallenge wiki on Antihydra https://wiki.bbchallenge.org/wiki/Antihydra ; Quanta, Aug 2025 https://www.quantamagazine.org/busy-beaver-hunters-reach-numbers-that-overwhelm-ordinary-math-20250822/ ; Mahler's 1968 paper (reprint) https://ems.press/books/dms/252/4994
Tag
number-theory
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