One of the most interesting patterns that has ever been constructed in Conway’s Game of Life is primer, a gun that fires lightweight spaceships

Nathaniel Johnston » Generating Sequences of Primes in Conway's Game of Life

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2022-01-17 18:00:15

One of the most interesting patterns that has ever been constructed in Conway’s Game of Life is primer, a gun that fires lightweight spaceships that represent exactly the prime numbers. It was constructed by Dean Hickerson way back in 1991, yet arguably no pattern since then has been constructed that’s as interesting. It seems somewhat counter-intuitive at first that the prime numbers, which seem somehow “random” or “unpredictable”, can be generated by this (relatively simple) pattern in the completely deterministic Game of Life.

The gun works by firing lightweight spaceships westward, and destroying them via glider guns that emulate the Sieve of Eratosthenes. A lightweight spaceship makes it past the left edge of the gun at generation 120N if and only if N is a prime number (though for technical reasons, 2 and 3 are not outputted).

It wasn’t too long after making primer that Hickerson realized that he could attach a gun to the bottom-left corner of it to turn it into a twin prime calculator by allowing each lightweight spaceship through only if another lightweight spaceship passed through 240 generations earlier. Similarly, Jason Summers constructed a Fermat prime calculator in 2000 by shooting a glider at the lightweight spaceship stream every generation of the form 120(2N + 1), which ends up detecting exactly the Fermat primes.