Jump to content
Sign in to follow this  

The Hidden DNA of Prime Numbers

Recommended Posts

Hey folks,

Here is possibly a neat "pattern" I've come across when studying prime numbers, or at least a different way of bucketing them. I'm looking to see if anyone can help me explain it, because I'm having a hard time wrapping my head around it. It may be that I've found something that is trivially explained away by some known information I just don't have or am not seeing. The pattern emerges with you cut prime sieves of length N > 3 into segments of 6 after separating the first 3 prime numbers (1,2 and 3). I'll explain below.

We'll be working with a prime sieves with the following properties:

  • We sieve on intervals [1, N] where N > 3. The examples work out nicely if N-3 % 6 = 0, 
  • In this sieve we'll represent a prime number at index 'i' with digit '1', and a composite number with digit '0'
  • We end up creating a string of '0's and '1's of length N that represents the primality of the number located at index 'i'.

Here is a sieve up to N=45, first separated by a segment of length 3 since 1, 2, and 3 are prime, and then subsequent segments of length 6.

 111 - 010100 - 010100 - 010100 - 010000 - 010100 - 000100 - 010100 ...

I've created these sieves all the way up to the 1 millionth prime number. The interesting thing that emerges is there are only 4 unique segments that ever show up:

  • 010100 - Segment that includes a twin prime (position 2 and 4).
  • 010000 - Segment that includes a single prime at position 2.
  • 000100 - Segment that includes a single prime at position 4.
  • 000000 - Segment that includes no primes at all.

There is never a case where the number represented at the 6th position in a segment is prime, EVEN THOUGH this digit position always represents an odd integer. There seems to be something interesting about grouping by 6. Even more interesting is that as N gets larger, it seems that the distribution of "010000" and "000100" seems to get closer and closer to being equivalent (approximately 16% when sieved up to the millionth prime)

The reason I'm calling this the hidden "DNA" of prime numbers, is because of the similarity to DNA in biology, i.e the four letters ACGT. There are four "letters" that emerge. It's even more interesting to encode the patterns into actual letters and then view the "strands". It'd be interesting to find patterns in the way the segments group together and if there are emergent rules to the chaos.

I have jupyter notebooks full of related information to this stuff if anyone is interested. Otherwise, I'm keen on hearing other input into what I'm looking at.




Edited by dthomas

Share this post

Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now
Sign in to follow this  

  • Create New...

Important Information

We have placed cookies on your device to help make this website better. You can adjust your cookie settings, otherwise we'll assume you're okay to continue.