I was reading Martin Gardner's Mathematical Games column on the Ulam spiral which appeared in the March 1964 issue of Scientific American. (The spiral actually featured on the cover of that issue.) Gardner makes the following statement:
The grid on the cover suggests that throughout the entire number series expressions of this form are likely to vary markedly from those "poor" in primes to those that are "rich," and that on the rich lines an unusual amount of clumping occurs.
By "this form" Gardner means the form $4x^2+bx+c$. I'm curious – and a little bit skeptical – about his last statement concerning clumping. I know that the existence of prime-rich and prime-poor polynomials is a longstanding conjecture, going back to Euler's discovery that polynomials such as $x^2-x+41$ generate unusually many primes, and that Hardy and Littlewood and also Bateman and Horn made concrete proposals as to what the density of primes in such polynomials ought to be.
My question is whether there is any evidence, either numerical or heuristic, that there should be a large amount of clumping in the primes of the form $x^2-x+41$. Famously, the first 40 values of $x$ all give primes, but if one goes to higher values of $x$ are there more long clusters of primes than one would expect if the primes were randomly distributed?
Rephrasing the question: I am aware of the conjecture that $x^2-x+41$ has more primes than other, similar lines. The question is whether there is a conjecture saying that $x^2-x+41$ has more dense clusters of primes than expected.
Best Answer
You will find this paper of interest: