Help Needed

[Jonathan Post]

Coloring the square and the hexagonal spiral by semiprimes is the basis of
several sequences of mine.See, for example:

A113688
A113653

and the references therein.

**



On 1/2/07, David Wilson <davidwwilson at comcast.net> wrote:
>
>  The profusion of primes of certain polynomial forms has been known for a
> very long time.
>
> The primes and semiprimes appear to from lines with upward and downward
> slope of varying density emanating in both diagonal directions from pixels
> with number p in the top row. Those emanating ot the left represent numbers
> of the form r^2 - r + p while those to the right represent numbers of the
> form r^2 + r + p, which are in fact the same numbers.
>
> The lines with the most prime pixels emanate from pixels p in the set
> {2,3,5,11,17,41}, which are known as the Euler Lucky numbers. The solidity
> of the lines emanating from pixel p are related to the class number of the
> complex quadratic field Q(sqrt(1-4p)). In the case of the Euler Lucky
> numbers p, 1-4p is a Heegner number and Q(sqrt(1-4p)) has class number 1,
> which is to say it is a unique factorization domain.
>
> Certain other polynomials, such as 2x^2 + 29, seem to be
> unusually impregnated with primes. I have no idea if these polynomials
> have a similar explanation to the ones you observe.
>
> Linear visual patterns of primes associated with quadratic polynomials
> such as the one you observe were noted by Stan Ulam in his prime spiral back
> in 1963 (which see at MathWorld). Ultimately, Ulam was unable to explain
> much of what he observed.
>
> The idea of coloring the prime spiral pixels according to almost-primality
> seems new though, and might produce some interesting pictures.
>
>
>
>
>
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