The Cookson Wheel: A Prime Number Pattern Discovery?

Well, this is interesting.

Earlier today, I ran across a reference to "the Ulam Spiral," a prime-number graphing system which was discovered/invented by Stanislaw Ulam, one of the key physicists of the Manhattan Project, while goofing off and doodling on scratch paper during a math conference in 1963. I emailed the link to a friend, Eric Cookson (trivia: his father essentially invented the DVD), and Eric seems to have, um, improved upon Ulam's work.

First, some background:

While doodling, Ulam started graphing the natural numbers in a spiraling concentric square pattern like so:



He then circled all the primes, which resulted in a pattern that looks like this, where the primes seem to line up in readily detectable diagonal patterns:



As the Wikipedia entry on Ulam Spiral explains:

All prime numbers, except 2 and 5, are odd numbers that end with 1, 3, 7, or 9. Since in the Ulam spiral adjacent diagonals are alternatively odd and even numbers, it is no surprise that all prime numbers lie in alternate diagonals of the Ulam spiral. What is startling is the tendency of prime numbers to lie on some diagonals more than others, while a random distribution is expected.

Tests so far confirm that there are diagonal lines even when very large amounts of numbers are plotted. The pattern also seems to appear even if the number at the center is not 1 (and can, in fact, be much larger than 1). This implies that there are many integer constants b and c such that the function:

f(n) = 4n2 + bn + c

generates an unexpectedly-large number of primes as n counts up {1, 2, 3, ...}. This finding was so significant that the Ulam spiral appeared on the cover of Scientific American in March 1964.

At sufficient distance from the centre, horizontal and vertical lines are also clearly visible.

Here is the "big" version of the Ulam Spiral, where you can see some of the diagonal patterns quite clearly:



All of the above is prefatory to what my friend Eric Cookson seems to have stumbled upon, after I sent him the link about Ulam Spiral (sending links to stories like that to each other is the sort o thing that the +4SD crowd does and is more commonly known as "The Road to Nerdition").

Eric saw that if you arranged the natural postive integers in a 12-radius wheel, a very striking pattern of prime numbers quickly and relentlessly emerges. Check this shizznit out:



Here is how Eric describes the phenomenon in his emails:

So far this is what I’ve gleaned... A number X may be a prime if

12x+1
12x+5
12x+7
or
12x+11

...is not divisible by 5, 7, or 11.

Which has worked for all numbers I’ve tested so far...well hey, it could just be dumb luck.

The only thing I seem to be able to figure out is that it eliminates 2/3 of the numbers as prime-possibles right off the bat...the further elimination seems to have some hitches...needs work and refinement, but I’d love to hear his [Derbyshire's]thoughts on it.

And he ends with this intriguing thought:

What would be really interesting is to see what happens when all the prime numbers line up on the same spoke, just need to figure out how many divisions I need around the circle, since they’re only on 4 of 12 spokes right now...hmmm.

Now here's the eerie part: the pattern that intuitively occurred to Eric is quite similar to another pattern, one central to Ulam's life -- who was described by Hans Bethe as "the father [of the H-Bomb], because he provided the seed, and Teller is the mother, because he remained with the child. As for me, I guess I am the midwife."

The universal warning symbol for Ulam's child is the following:



Look familiar? Cue Twilight Zone music...

If you want to discuss this with Eric, email me at wordwarp (AT) ay oh el (DAHT) com and I will pass it along to him. Actually, maybe I'll just turn comments back on.

UPDATE: Eric writes,

Interestingly, as I examine the spokes more, I think it’s nothing more than a curiosity, which can be explained by divisibility along the spokes. The columns that are most easily eliminated are due to their relationships to the numbers 2, 3, or 4 (all possible prime factors for the 12)...no big deal, but neat nonetheless. The really curious thing still lies in how things skip along the spokes.