If you can explain them please enlighten us all. You will have to show that Hardy-Littlewood's Conjecture F is true. http://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture

 I'm not a mathematician, but I don't think you can deduce that conjecture from the Ulam Spiral. On the other hand, you don't need this conjecture to show e.g. 4x^2-2x+41 is rich in primes (within the limits of the picture). If I recall, you can show the clumping arises from rewriting the primes in base 6.EDIT: To hurry along the conversation a bit - it's entirely possible I've been misled into thinking things are more simple than they really are. Wouldn't be the first time...
 Sorry if I was a bit snarky in my first reply. As far as I understand this, you can't explain the structure of the Ulam spiral in a trivial way. You are correct that there is a trivial part to the spiral: diagonal lines alternate between odd and even numbers, therefore all the primes lie along diagonal lines. However, the structure is much richer than that.The question is why certain lines have lots of primes and not others? why is a given polynomial so rich in primes, while other similar ones are not? That's what some of these conjectures are trying to prove.
 diagonal lines alternate between odd and even numbers, therefore all the primes lie along diagonal linesHuh? If you draw the Ulam spiral on a checkerboard, all odd numbers end up on squares of one color and all even numbers of squares of the other color, but that is neither necessary nor sufficient to get those diagonal streaks in the picture.

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