Fermat prime calculator
|Fermat prime calculator|
|View static image|
|Number of cells||10001|
|Bounding box||838 × 736|
|Discovered by||Jason Summers|
|Year of discovery||2000|
The Fermat prime calculator is a pattern that was constructed by Jason Summers on January 7, 2000 that calculates Fermat prime numbers. In particular, it is rigged to self-destruct and stop growing if any Fermat primes over 65537 are found. Since the existence of such primes is an unsolved problem in mathematics, it is unknown if this pattern exhibits infinite growth.
The tubs that are arranged in a diagonal line at the bottom-left corner of the pattern represent the known Fermat primes, starting with 5. The other three represent 17, 257, and 65537. A tub gets destroyed whenever a Fermat prime is found. In the unlikely event that an additional Fermat prime exists, the nearby pond will eventually be destroyed, followed by the beehive puffers that border the pattern and then the breeders that make up the rest of the pattern.
A number N is tested for Fermat primality at about generation 120N - 550, so the tubs will be destroyed at about generations 50, 1490, 30290 and 7863890. The next smallest Fermat number that is not known to be composite is F33, or 22^33 + 1. Therefore, the pond (and the breeders) will survive at least 102580000000 generations, so don't expect to see anything interesting happen by watching this pattern. Remove the tubs manually if you want to see the pattern self-destruct.
The breeders each advance at c/2. Since it is impossible for a spaceship to propagate through vacuum faster than c/2, the beehive fuses are required. The small arrangement of still lifes, including the pond, triggers the beehive fuses when hit by a glider. The beehive fuses are consumed at 4c/5, hence the disturbances overtake the breeders and destroy them.
The careful observer may notice that actually every number of the form 2N + 1 is tested for primality, while Fermat numbers are of the form 22^N + 1. However, numbers of the form MN + 1 can never be prime unless N is a power of 2, so the only primes of the form 2N + 1 are in fact the Fermat primes.
The Fermat prime calculator is based on the original prime number calculator, primer, and a caber tosser, both of which were found by Dean Hickerson. The beehive puffer was constructed by Hartmut Holzwart.
- This pattern's initial population is a fermat prime in binary (0b10001 = 17).
- fermat-primes.rle from Golly's built-in pattern collection
- Nathaniel Johnston, Dave Greene. Conway's Game of Life: Mathematics and Construction (2022), 8.8.4. Guns and Glider Streams: Fermat Primes and Unknown Growth