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.