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<DIV><FONT face=Arial size=2>The profusion of primes of certain polynomial forms
has been known for a very long time.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>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 </FONT><FONT face=Arial size=2>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.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>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.</FONT></DIV>
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<DIV><FONT face=Arial size=2>Certain other polynomials, such as 2x^2 + 29, seem
to be unusually impregnated with primes.</FONT> <FONT face=Arial
size=2>I have no idea if these polynomials have a similar explanation to the
ones you observe.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>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.</FONT></DIV>
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<DIV><FONT face=Arial size=2>The idea of coloring the prime spiral pixels
according to almost-primality seems new though, and might produce
some interesting pictures.</FONT></DIV>
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