The 6-state Busy Beaver number BB(6) lower bound has been improved sequentially from over 10^36,534 to 10 tetrated 15 times, then to 10 tetrated 10 million times, and most recently to 2 tetrated to 2 tetrated to 2 tetrated to 9 (equivalent to 2 pentated to 5).
The exact value of BB(5) is known as 47,176,870, highlighting the dramatic growth leap between n=5 and n=6.
To convey the vastness of the new BB(6) bound, one can imagine 10 tetrated 10 million grains of sand filling that many observable universes.
These breakthroughs hint that the Busy Beaver function might become independent of the ZFC axioms at much smaller n (around 7 or 8) than the previously proven threshold of n=643.
The latest improvements come with formal correctness proofs in Coq and have sparked community discussions on the specific Turing machines and the nature of their nonhalting or halting proofs.
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