So, in entertaining passing fancies about primes, I had an amusing thought for building an analog primality tester.
Imagine a device that is two parallel panes of glass separated by a small distance, like an ant farm. The floor is rigid and flat, and two orthogonal sides are rigid and straight, but can be slowly cranked towards or away from each other.
At any set distance (of the sides), we begin a primality test for the number N by filling the device with N identical balls of diameter d. The parallel panes of glass are just a smidge more than "d" apart. The balls at the bottom fill the device and make a rectangle. At any point while cranking the sides together and apart, the top row may, or may NOT have, enough balls to make the top row "complete" - i.e., it may have fewer balls than the rows below it. S
To complete the test one must crank the sides slowly from close together (a smidge > 2d) to very far apart (just > dN/2). If, at any point the top row is complete, making a complete rectangle -- voilà, there are your factors and N is composite.
1. Has anyone ever seen this realized physically? I haven't ever seen it realized in hardware. Its pretty brute force, but it does employ massive parallelism.
2. Assuming hydrogen atoms for balls, of Bohr radius .53 angstroms, how big a device would we need to test 2^57,885,161-1 for primality.