Exploring possible interfaces between primes (analytic number theory) and the space of CA rulesets.
June 23, 2017—Daniel Reynolds
What Does It Look Like When We Visualize the Spacing between Consecutive Prime Numbers?
When looking at prime numbers, one of the distinctive patterns is the spacing between consecutive primes. In order to get the spacing, we can write a function that takes the differences between n consecutive primes in a nested list corresponding to a finite difference table of absolute differences.
Now we can look at one of these nested lists for the first 25 primes. Just looking at the raw data, we see that after a certain number of steps, all the data turns to being only 2s and 0s except for a constant 1 at the start of each list step.
When we create an array plot of the first 200 primes and their different steps of data, where each of the zeros is turned to black and other numbers (mostly twos) become gray, we see an image that looks very much like a cellular automaton. The black triangles are clusters of zeros in the difference table corresponding to our array plot.
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This pattern is a modern visualization of something recorded as discovered and published by François Proth in 1878 and Norman L. Gilbreath in 1958. It is usually called Gilbreath’s conjecture, where the first entries in each step are conjectured to follow the series 2,1,1,1,1,1,1,1....
Is This Patterning an Artifact Particular to Spacing in Prime Fields?
Is There a CA Ruleset That Corresponds to the Spacing between Consecutive Prime Numbers?
FURTHER EXPLORATIONS
Could Any Further Interfaces between CAs and Prime Numbers Be Discovered?