Meet the Busy Beaver number, a number so huge that mathematicians call it the frontier of mathematical knowledge
Mathematicians have discovered a number so large it’s at the edge of what we can know.
The Busy Beaver problem is a mind-bending but fascinating idea from computer science and mathematics that asks: what is the longest possible time a simple computer can run before it stops, if it only has a limited number of instructions?
These tiny computers are called Turing machines, and in this case, mathematicians have turned their attention to ones with just a few states and a tape with two possible symbols.
For each number of states (instructions), there are many possible machines, and the Busy Beaver number — first formulated by the Hungarian mathematician Tibor Radó in 1962 and written as BB(n) — is the maximum number of steps any of them can take before stopping. You try every combination and pick the one that runs the longest—but still eventually stops.
For example, BB(1) is just 1 step. BB(2) is 6 steps. BB(3) runs 21 steps. BB(4) jumps to 107 steps. These numbers sound small, but the growth, as you can probably tell, is extremely fast.
In 2024, BB(5) was finally solved after decades of painstakingly sifting through 17 trillion possible Turing machines. The best-performing machine ran for over 47 million steps before halting—an enormous leap from previous values. And that’s still just with five instructions.
I don't understand...
OK, here is the simplest case: the machine only has 1 state, which you can think of it like a single line of code, or an instruction, it can follow. There are exactly 25 possible machines in this setup.
Each machine will either stop right away or keep running forever.
The one that lasts the longest before stopping runs for just 1 step.
Therefore, BB(1) = 1.
If we do the same for 2 instructions, we get: BB(2) = 6
And so on...
BB(3) = 21
BB(4) = 107
BB(5) = over 47 million!
Now, researchers are turning their attention to BB(6), and things get absolutely wild. The number of possible machines to check explodes to around 60 quadrillion, which would take vast computing power to process. But even more astonishing is how many steps the best-performing machine takes.
We don’t know the exact number yet, but experts have found some lower bounds—and they’re huge. So huge, in fact, that we can’t even write them using normal exponents.
Mathematicians can’t even write the number down using regular math like “millions” or “billions” or even powers like “10 to the 100.” Instead, they use something called tetration, which is like stacking exponents on top of each other. Imagine doing 10 to the power of 10 to the power of 10… 15 times: BB(6) is bigger than 10^10^10^... 15 times, and is likely bigger than the number of atoms in the universe.
A number so big it nearly breaks mathematics!
— New Scientist (@newscientist) July 12, 2025
Busy Beaver numbers are used to describe the longest possible run-times of ‘Turing machines’.
But @jjaron says there's been a new development in the quest to figure them outhttps://t.co/kzzN2JTXez pic.twitter.com/jnaw3Yl4ql
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