Hi guys, I've been playing around with structure generated patterns and have come across one which has caught my attention. I am only just starting to learn about number theory and so I am sure someone might be able to provide an explanation for this. Let me describe what I did then show you the results:(adsbygoogle = window.adsbygoogle || []).push({});

The motion of the path is the same as a Ulam spiral. For those of you not familiar with this structure, you can read about it here:

http://mathworld.wolfram.com/PrimeSpiral.html

If you go to that link, you can observe the square spiral path the numbers follow. Notice that the path progresses by counting all the positive integers, starting with 1 (1,2,3,4...).

This is where I made my change. I figured that since half of the numbers on the path of a standard Ulam grid were even, and if all we wanted to do was color the prime numbers, then half of the space was being wasted. By only considering odd numbers on the path, blank spaces would have more 'meaning' from a visual point of view, since there was at least the possibility that the space could have been prime.

So I set the starting point to 0, and the from that point assigned successive ODD NUMBERS ONLY as the path progressed (0,1,3,5,7...). What I found was that there were two columns and two rows which contained absolutely NO primes (except for 7). I found this interesting, as I had verified this for up to around the first 250,000 primes.

I am attaching 3 pictures of the sequence, here are some things to keep in mind:

1) most examples of the Ulam spiral (like the one in the link provided) start at the center, move right, and then follow an anticlockwise pattern.

In my algorithm, I start at the center, move up, then follow a clockwise pattern. Note this just changes the paths orientation on the grid, it doesn't affect the pattern.

2) I refer to each full rotation around the center as a turn.

3) I highlighted the columns and rows I was talking about in green for ease of viewing them. the matlab file doesn't do that on its own.

the three pics are at 10, 50, and 250 turns. I am also attaching the matlab file I used(see next post). I did it up to turn 690 but it froze before I saved the results >.< the property still held tho. Feel free to play around with it if you like. there may be more interesting patterns that a more trained eye can see.

This is probably trivial, but I thought I should share =) Hopefully someone could explain how this works to me.

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# The Odd Only Spiral (A twist on the ulam spiral with an interesting property)

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