The night sky we see as a random scattering of stars is an aspect of the order of galaxies. Primes, scattered through the whole numbers, are patterns superimposed on patterns, obscured by multiplicity like the faint path of the Milky Way.
How does it work?
The pattern appears because as you convert the numbers to each base, certain areas are more likely to be odd or even; the even numbers cannot be prime. The simplest way to see is to think of the base 10 example: primes between 100 and 200 when reversed end with a 1, and so are more likely to be prime; between 200 and 300 reversed end with a 2 and cannot be prime. Different bases become a bit more complex.
5 converted to base 2 is 101. Reversed, this is 101 again, or 5 in base 10, which is still prime. In base 3, 5 is 12. Reversed, it is 21, or 7 in base 10, which is still prime. 5 in base 4 is 11, which reversed is 11, or 5, so still prime. So 5 appears as 5: # # # .
|in base||2||3||4||5||and so on|
Can this pattern be used to generate large primes?
In general, mathematicians are unconcerned with palindromes and string-type readings of numbers. However, it is relatively simple to convert a number to another base. Since the bands of prime intensity are in regular areas, it's possible to use trigonometry to find the bases to convert to (since you know the length and adjacent angle.) This doesn't guarantee the new number generated will be prime, but makes it more likely. But while the new number may be prime, but not a larger prime than the original.
An infinite array
Questions: Is this pattern related to prime-generating polynomials? Are Mersenne primes specially represented here? Why are some primes more radiant than others, for example 157? Why do the empty areas produce wave like patterns or whorls?
The Primes and Beyond
An image of the message sent to M13 from Arecibo encoded with primes:
Clarke and the Primes
"Jeserac sat motionless within a whirlpool of numbers. The first thousand primes..."
Arthur C. Clarke wrote about the glamor of primes in his novel, The City and the Stars. His description anticipates the prime spiral discovered by Ulam. See the Mathworld link above for a description and the complete quote.