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Comment by BlackFly | Hacker News Reader

Isn't the answer to "Why do they cluster on diagonals?" Really mundane? The answer of course being: horizontal and vertical lines are forbidden because all non-diagonal neighbours of an odd number are even. Then your mind focuses on all the runs because that is what the human brain does.

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Here is a quick experiment to show that the Ulam spiral indeed appears to have more prominent diagonal lines than a spiral of randomly chosen odd numbers even when the dimensions and the number of points in both spirals are same: https://github.com/mycask/ulam-vs-random

I have plotted a 99x99 grid in this experiment. One can plot larger grids to see that the difference between the two grids (the Ulam spiral and the random spiral) becomes more apparent with larger sizes.

Update: After I made these text-plots, I found that Wikipedia has much better graphical plots that demonstrates the difference:

- Ulam spiral: https://en.wikipedia.org/wiki/File:Ulam_1.png

- Random spiral: https://en.wikipedia.org/wiki/File:Randomly_black_odd_number...

> 200x200 spiral number grid with odd numbers that have a 23.38% chance of being coloured in black.

This doesn't seem to take into account that big prime numbers are less "likely" than small ones.

Yes, if you take into account all factors that might cause diagonal streaks more likely on real spiral as opposed to random grid, you might actually find out the real cause of this strange fact.

IIRC the pattern still arises even if you generate pseudo-primes randomly.

The square root plays a role in the distribution of primes as well as in the area of a circle. The former is involved in an upper bound for the numbers to be checked performing Sieve of Eratosthenes.

Here's another page who does this.

http://ulamspiral.com/generatePage.asp?ID=1

It's not just the diagonal runs but the dense clusters along some diagonal lines that is considered to be an unsolved problem.

Moreover, the clustering is not limited to just diagonal lines. There are less prominent but high density clusters along horizontal and vertical lines too.

Nope there is a real pattern -- they cluster along certain polynomials.

Trying picking random points on the black squares of a chessboard (same condition that you stated), they won't make obvious lines like that.

Couldn't you simulate this pretty easily? You would need some kind of objective way to measure "diagonal clustering", but you could try to simulate numbers several thousands of times and then see if the observed amount of diagonal clustering is matched in some non-trivial fraction of the simulations.

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