Help Needed

[David Wilson]

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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