Prime Atolls
Posted: Sat Mar 07, 2026 6:52 am
Around 2000 I found that if the natural numbers are arrayed in prime-number columns, the prime numbers align in diagonals (I got the idea after reading about the Ulam Spiral). Certain prime numbers, such as 11, 13, 19 and 23, produce diagonal oriented rectangles which I call prime atolls, because they resemble atolls of primes in a sea of compound numbers. They are usually partially submerged too, the submerged parts always being semi-primes.
They have fascinated me ever since and have interesting properties. For example, the twin primes are always found horizontally across from each other within an atoll, for all the column numbers I've tried.
I noticed today that in 19-atolls the absolute primes are always aligned along the same top-right diagonal, and can be identified using modular arithmetic. So you can see 13/31, 37/73, 79/97, 113/131/311, 337/373/733 all on the same diagonal. I've only shown the first 600-odd natural numbers, but it's easy to calculate which diagonal a number is on by calculating the residue, modulo-18.
So 337 and 373 are on diagonal 11, because ((337/18)-18) x 18 = 11, and ((373/18)-18) x 18 = 11. So we can find out if 733 is also on the 11-diagonal without extending the rows, by the same calculation.
((733/18)-18) x 18 = 11,
Therefore it's also on the same diagonal.
It's the same with 199, 919 and 991, which are all on diagonal 1.
1. The sum of each atoll (including the semi-primes) is always 240 or a multiple thereof.
2. The largest number in any atoll is aways 74 more than the smallest.
3. Numbers increase by 18 down the top-right diagonals
4. Numbers increase by 20 down the top-left diagonals.
5. The gaps in the atolls are always semi-primes.
6. The submerged numbers along the centers of each atoll always have 3 prime factors.
7. The most highly compound numbers are between the atolls.
Note that 2 and 5 do not belong to any atoll here. The first few prime numbers (2, 3, 5, 7) often stand alone, depending on the column width.
Anyway, I thought some of you may find it of interest. There are lots of questions they give rise to, such as
1. How many complete atolls are there for each type?
2. How many prime numbers are associated with atolls? Some seems to just give diagonals, but it may be that they are particularly long.
3. What's the first completely submerged atoll for each type?
4.Do other kinds of prime numbers show patterned behavior within these atolls?
Bill
They have fascinated me ever since and have interesting properties. For example, the twin primes are always found horizontally across from each other within an atoll, for all the column numbers I've tried.
I noticed today that in 19-atolls the absolute primes are always aligned along the same top-right diagonal, and can be identified using modular arithmetic. So you can see 13/31, 37/73, 79/97, 113/131/311, 337/373/733 all on the same diagonal. I've only shown the first 600-odd natural numbers, but it's easy to calculate which diagonal a number is on by calculating the residue, modulo-18.
So 337 and 373 are on diagonal 11, because ((337/18)-18) x 18 = 11, and ((373/18)-18) x 18 = 11. So we can find out if 733 is also on the 11-diagonal without extending the rows, by the same calculation.
((733/18)-18) x 18 = 11,
Therefore it's also on the same diagonal.
It's the same with 199, 919 and 991, which are all on diagonal 1.
- 19-Atolla.png (64.99 KiB) Viewed 1945 times
1. The sum of each atoll (including the semi-primes) is always 240 or a multiple thereof.
2. The largest number in any atoll is aways 74 more than the smallest.
3. Numbers increase by 18 down the top-right diagonals
4. Numbers increase by 20 down the top-left diagonals.
5. The gaps in the atolls are always semi-primes.
6. The submerged numbers along the centers of each atoll always have 3 prime factors.
7. The most highly compound numbers are between the atolls.
Note that 2 and 5 do not belong to any atoll here. The first few prime numbers (2, 3, 5, 7) often stand alone, depending on the column width.
Anyway, I thought some of you may find it of interest. There are lots of questions they give rise to, such as
1. How many complete atolls are there for each type?
2. How many prime numbers are associated with atolls? Some seems to just give diagonals, but it may be that they are particularly long.
3. What's the first completely submerged atoll for each type?
4.Do other kinds of prime numbers show patterned behavior within these atolls?
Bill