# Fermat prime calculator

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.

Fermat prime calculator

x = 838, y = 736, rule = B3/S23 46b3o12b3o\$46bo2bo10bo2bo\$46bo16bo\$46bo3bo8bo3bo\$46bo6b4o6bo\$47bobobo 2b2o2bobobo\$51bo6bo\$51bo6bo\$50bob2o2b2obo\$50bo3b2o3bo\$51bo6bo\$51b2o4b 2o\$54b2o\$49bo3bo2bo3bo\$53bo2bo\$50b4o2b4o\$52b2o2b2o\$53b4o\$52bo4bo2\$52bo b2obo\$51bo6bo\$51bo6bo\$53b4o\$49b2obob2obob2o\$48bo2b2ob2ob2o2bo\$47b2o2b 8o2b2o\$47bo3bo2b2o2bo3bo\$46b2o5bo2bo5b2o\$46b2o3b3o2b3o3b2o\$47bo2b2obo 2bob2o2bo\$48b2obob4obob2o\$53bo2bo\$48bo12bo82b3o15b3o\$48bo4bo2bo4bo81bo 3bo13bo3bo\$50b2o6b2o82b2o4bo11bo4b2o\$44bo5b3o4b3o81bobob2ob2o3b3o3b2ob 2obobo\$43b3o8b2o84b2obo4bob2ob3ob2obo4bob2o\$43bob2o4bobo2bobo75bo4bo4b o3bo4bobo4bo3bo4bo\$44b3o2bob2o4b2obo69b2ob3o15bo5bo\$39b3o2b3o2bo10bo 67b2obo3b2o2b2o7b2o9b2o7b2o\$38bo2bo4bo77b5o4bo2b2o\$41bo9bo6bo61b2ob2ob obobo5bo2bo\$41bo4b3o4b4o7b3o15b3o33b2obo3bo3bo8bo\$38bobo4b3o15bo3bo13b o3bo28b5o4bo\$45b3o14b2o4bo11bo4b2o23b2ob2obobobo3bo\$48bo5b2o5bobob2ob 2o3b3o3b2ob2obobo20b2obo3bo3bo\$48b2o3bo2bo3b2obo4bob2ob3ob2obo4bob2o 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Pattern type Miscellaneous
Number of cells 10001
Bounding box 838×736
Discovered by Jason Summers
Year of discovery 2000

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.[1]

 The tubs that are destroyed by the discovered Fermat primes The calculator self-destructs and stops growing if the tubs are removed

## Videos

 The Fermat prime calculator destroying the tubs representing the Fermat primes 5, 17, and 257

## References

1. fermat-primes.rle from Golly's built-in pattern catalogue. Retrieved on July 28, 2009.
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