All the twin primes (primes only 2 apart, like 11/13, 41/43, etc) are found together on the 11-grid, as they are on other grids. They sit horizontally across from each other on the atoll. The maximum possible pairs is 3 per atoll.

Not every grid with a prime number of columns forms atolls. So far I've found them in grids 11, 19, 23 and 29 columns wide. 7, 13, 17, 31 and 37 columns results in simple diagonal lines of primes, like the Ulam Spiral. The atolls can slope in either direction and can consist of more than eight numbers. For example 29-atolls are six numbers long and two wide. Each 29-atoll is a multiple of 504.

It's of little use, but since all primes must belong to an atoll, and since the position of atolls on grids follows a regular pattern - for 11-atolls repeating every 21 rows - I believe that in a third of cases it can be shown without a lot of calculations that a very large number is not prime, just by calculating its position on, say, an 11-grid. If it lies between atolls it can't be prime.

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