Thread: Prime clusters

I was exploring what I call prime clusters which leads to a puzzle.

If you check successive square numbers, you'll find two prime numbers (2 and 3) between 1 and 4, two prime numbers (5 and 7) are between 4 and 9, two prime numbers are between 9 and 16, three prime numbers are between 16 and 25...

Here's a more extensive list where the first number ahead of the colon is the second square number (from the range) and the number after the colon says how many prime numbers lie within that range. So we have:

4:2 9:2 16:2 25:3 36:2 49:4 64:3 81:4 100:3 121:5 144:4 169:5 196:5 225:4 256:6 289:7 324:5 361:6 400:6 441:7 484:7 529:7 576:6 625:9 676:8 729:7 784:8 841:9 900:8 961:8 1024:10

Removing the square numbers (to improve readability), we have:

2 2 2 3 2 4 3 4 3 5 4 5 5 4 6 7 5 6 6 7 7 7 6 9 8 7 8 9 8 8 10 and it's evident that this sequence of prime clusters is steadily growing, starting from 2 and going up to 10 where I had stopped.

If I had used triangle numbers (starting from three), this is what I get for the prime cluster sequence:

1 1 1 2 2 1 2 3 2 2 3 3 3 3 2 4 3 3 4 4 4 4 4 4 4 4 5 5 6 4 5 3 6 6 7 5 5 6 4 8 5 6 7 8 6

Both sequences show steady growth (although I would comment that the string of eight 4's in a row is notable). The puzzle is to determine if the prime cluster number is unlimited in growth as the square or triangle number increases or if the prime cluster number doesn't exceed some finite number in magnitude, no matter how far you go with either the square numbers of the triangle numbers.

Good luck with this one friend.
The only thing I would comment on is your comment about the string of eight 4's in a row being notable.
I don't think it is too notable; as you said "no matter how far you go"- there is a long way to go and you may notice many patterns which may or may not be of significance.

Originally Posted by ark600

Good luck with this one friend.
The only thing I would comment on is your comment about the string of eight 4's in a row being notable.
I don't think it is too notable; as you said "no matter how far you go"- there is a long way to go and you may notice many patterns which may or may not be of significance.

However I have found from experience that it is unusual to get a string of primes greater than four (however again there are different strings depending on exactly what you're looking at).

I'm aware of Euler's prime generating polynomial, , and Wolfram has more to say on this. I think my comment about the sequence gradually increasing, especially the second sequence, is the most interesting part to my post because it seems to be linearly increasing (lacking a home computer, I'm unable to plot this out on a graph to see what happens as the sequence is extended further).

I don't know how deeply prime numbers have been explored. Other things come to mind. Regarding what I've posted, I do believe that both sequences are tending to infinity and I'm awaiting for someone to solve the puzzle.