We got a cubic polynomial which is unexpectedly prime rich.

Let $f(x)=29160 x^3 + 30132 x^2 + 8046 x + 643$ and

$\pi_f(n)$ the number of primes values of $f(x)$ for $x \in [1,n]$.

Let $F(n)=\frac{\pi_f(n)}{\frac{n}{\log{n}}}$.

$F(n)$ is greater than one on $F(10^n)$ experimentally increasing for $n \ge 4$.

Here is some data

```
n \pi_f(n) F(n)
10^3 200 1.381551055796
10^4 1455 1.340104524122
10^5 11772 1.355301585736
10^6 100279 1.385405583242
10^7 877710 1.414701373380
10^8 7807058 1.43811322967 #pseudoprimes according to pari/gp
```

In the range $[10^{100},10^{100}+10^4]$ there are $28$ primes while $f(n)$ is prime $67$ times. (Independent verification will be appreciated).

Maybe this is just the law of small numbers, but $f$ appears more prime rich than the naturals, which surprises us.

Q1 How to explain this experimental data? (Especially more primes values in the large range).

Q2 What is the constant $C$ in Bateman–Horn conjecture?

If someone tests experimentally, for large numbers pseudoprimality tests are significantly faster than deterministic tests and will be accepted.

If $f$ were linear, congruences likely would explain this.

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