The Hidden DNA of Prime Numbers

[dthomas]

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 December 16, 2019 by dthomas

[uncool]

For the sixth position: the number there is always divisible by 3, as it can be written in the form 6n + 3 for some integer n.

For the distributions of 010000 and 000100 being equivalent: https://en.wikipedia.org/wiki/Prime_number_theorem#Prime_number_theorem_for_arithmetic_progressions

[dthomas]

Thank you @uncool, I didn't see that simple explanation for the 6th position being the multiple of 3 + 6k. Also thank you for the link!