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I just stumbled on a set of prime-generating polynomials of the form $$9 n^2-3 H n+H (H+1)/4$$ (where $H$ is a Heegner number $>11$), which generate the same number of distinct primes as their more familiar counterparts. There is some overlap in the primes generated by $n^2 - n + 41$ and $9 n^2-489 n+6683$ for example, but they are not all the same.

Are these of any interest, and are there other prime-generating polynomials that are related directly to the Heegner numbers? Also, why are the coefficients $9$ and $3$ involved?

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According to Wikipedia, Rabinowitz showed that $n^2 + n + p$ is prime for $n=0, \ldots, p-2$ iff $p = (1+H)/4$, $H$ a Heegner number. If you substitute $n = 3 m - (H+1)/2$, you get your polynomial $9 m^2 - 3 H m + H(H+1)/4$.

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  • $\begingroup$ true, but if I substitute $n$ for anything other than $3 m - (H+1)/2$, I don't get all prime values for $m=0 \ldots p-2$. Why these values in particular? $\endgroup$
    – martin
    Feb 2, 2016 at 6:26
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There is a way to generate an infinite number of prime generating polynomials with of course a fluctuating number of primes that was posted in the math stackexchange few months back.

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