Some while back I was reading some arguments that analog computers are currently underrated. Then not too long after that I came across this video about how we don’t know if π^π^π^π is an integer because the result would be so large so as to be intractable (even if we don’t care about the most significant digits and would modulus them). One of the benefits of analog computers is that they can solve problems near instantaneously compared to digital ones (I believe this is due to parallelism in most cases, although the speed at which electricity moves through wires is another). I also knew that they often mathematical calculations via the rotation of gears, which intuitively struck me as suitable both to modulus and operations involving π. There is the issue with how precise your machines parts are for that, and perhaps the error/noise would grow exponentially for that large exponential function. I gather slide rules are used to calculate exponentials, but (having never used one, or any analog computer) myself, my understanding is that they depend on having pre-calculated numbers on different sections to ensure the correct relation. Skepticism toward the ability of analog computers to efficiently solve problems difficult for digital ones is found in this paper from Scott Aaronson (which he discusses near the end of this blog post) and this one arguing for a “Strong Church’s Thesis” on the resources required for an analog computer to solve such problems. The thing is, those papers are talking about taking traditional problems given to digital computers using finite inputs and finite computational time and then translating that to an analog computer. π cannot actually be expressed finitely (unless you simply declare your base unit to be in terms of it), and as a transcendental number requires an infinite series to calculate. And this isn’t an issue of us searching a large number of possible solutions to see if any match a problem like with the Traveling Salesman, it’s a problem of arithmetic. I’m guessing the video is correct and it can’t be feasibly be done (even to an approximation) or someone would have done it, but I feel like I don’t know nearly enough about analog computing to know how the best way to go about such a calculation would be and why that wouldn’t work, so the premise that it couldn’t work still feels unproven to me.

April 23, 2021

## Could one calculate if π^π^π^π is (nearly) an integer with an analog computer?

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