Prime Atolls

[thebluetriangle]

Hi everyone.

In 2000 I made what I thought was an interesting discovery about prime numbers. I got very excited about it for a while then forgot about it. But I thought I'd post it here for your comments. It's been proven that there is no real pattern to prime numbers and therefore no way of predicting whether or not a very large number is prime. However, there are partial patterns that are interesting to contemplate and which can reduce the time required to factorise very large numbers. For example, other than 2, no prime is even, and other than 5 itself, no number ending in 5 is prime. Other than 2 and 5, all primes end in 1, 3, 7 and 9.

One of the more startling patterns is the straight lines prime numbers make on the Ulam Spiral, a spiralling of the natural numbers out from the number 2 like a Catherine Wheel. Mathematician Stanislaw Ulam discovered it while doodling during a boring lecture (although others had apparently found it before him). After reading about the Ulam Spiral I had the idea to put the natural numbers in a rectangular grid a prime number of columns wide. So for example one grid has eleven columns, another 17 columns, etc. When this is done, the prime numbers often arrange themselves in diagonally oriented rectangles. Here is the grid 11 columns wide with the prime numbers highlighted.

Notice how the primes always cluster in discrete rectangular units. I call them Prime Atolls, because as with real atolls the 'sea' of composite numbers around them fills their centre and usually leaves them partially submerged. Note that I have coloured 2, 5 and 7 red. This is because they may or may not be part of an atoll depending on the number of columns.

Here are some of the other features for the atolls found in an 11-column grid.

1. Each atoll is formed from eight numbers.

2. With the exception of 2, 3 and 5, all of the prime numbers appear to form part of an atoll ring.

3. Only the first atoll is whole. All the others (I've checked up to 400) are partial.

3. The submerged parts of atolls are semi primes as far as I've checked (up to 400).

4. They sum to 240 or multiples thereof. So the nth atoll sums to 240n.

5. Each atoll is formed from 2 parallel rows of primes or semi-primes, which, reading top left to bottom right, always end 7, 9, 1, 3.

6. Each component of the nth atoll is 30 more than its equivalent on the (n-1)th atoll.

7. The largest number in an atoll is always 46 more than the smallest. Many similar observations can be made.


I can't see a use for prime atolls, so this likely falls within the realm of recreational mathematics. But there is always the remote possibility that a new property of numbers is in there to be discovered. The Ulam Spiral itself has been studied by many mathematicians, so prime atolls may also be deserving of some attention. At any rate I find them interesting to contemplate. Many questions can be asked of them. For example:

1. What is the first totally submerged atoll for 11 columns? By inspection of the 11-atolls, the gap would have to be at least 66 (31 to 97, for example). There is a gap of 71 between the primes 31397 and 31469, but because of the grid position of the atoll beginning at 31397, more than a gap of 66 is required and there isn't enough room for a complete atoll. I believe the first one may be between 155921 and 156007. It astonishes me that we have to go so far down the number line to reach the first totally submerged atoll (one containing no primes).

2. Are the atolls composed solely of primes and semi-primes? I would suggest that the answer is no, but I haven't found an example yet for 11-atolls of a number with three or more factors (not including 1).

I don't have enough knowledge of mathematics to study them in much more depth, so if anyone else finds them as intriguing as I do and wants to do so, then be my guest. Maths isn't always important, but it should always be fun and I've had a lot of that from contemplating these unexpected patterns.

[thebluetriangle]

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.

[TerryMulhern]

Thanks for the information.
I've been just searching for the mathemathical tools in the theory of numbers developed by Stanislaw Ulam and found this thread. Not so long ago, I was digging around Atomic Heritage Foundation to find the info for the University of Colorado math project I'm working on.