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Comment by rgoddard | Hacker News Reader

You are going to see a pattern along the diagonal because every other number is even and never prime (excluding 2). So the only way for two primes to be touching is along a diagonal. Given that and our brains tendency to group items which are closer together as being related, anything along the diagonal will stand out more and appear to be a pattern. A more interesting picture would be one where all of the even numbers are omitted.

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A more interesting picture would be one where all of the even numbers are omitted.

I made canvas renderer to check it:

http://alteredqualia.com/visualization/ulam-spiral.html

It's not checkerboard, but there are horizontal and vertical patterns.

All the odds would look like a full checkerboard, not like these diagonals. The fact that some diagonal lines appear to be favored over other nearby diagonal lines of odd numbers is still interesting.

Not coloring all the odds, just plotting only odds and still coloring only the primes. The new spiral has a 3 where the old two was and a 5 where the old three was, etc.

Of course it's interesting. There's no such thing as an uninteresting number. http://en.wikipedia.org/wiki/Interesting_number_paradox

There's also the pattern that every prime (above 3) modulo 6 is congruent to 1 or 5. I suspect that some of the pattern comes from that fact.

Is there a more generalized version of that pattern, perhaps something like this (I haven't checked):

  let P(n) be the nth prime number: P(1)=2, etc.
  let n be an integer >= 2, and m be an integer > n
  then P(m) ≡ 1 and (P(n) - 1) (mod P(n))

?

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