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.



Unexpectedly prime rich cubic polynomial
The expected constant in the [Bateman-Horn conjecture](https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture) is 
$$\frac1d \prod\_p\frac{1-\frac{n\_p}{p}}{1-\frac1p},$$ where $n\_p$ is the number of roots of $f(x)$ modulo $p,$ and $d$ is the degree of $f(x).$ For the particular polynomial in question, this converges quite rapidly, and when the product is taken over the first 10000 primes, the constant is approximately $1.6235,$ which does not disagree with the experimental result.


For posterity, here is the *Mathematica* program:



```
f[x_] := 29160 x^3 + 30132 x^2 + 8046 x + 643
nn[p_] := Length[Solve[ff[x] == 0, x, Modulus -> p]]
rat[p_] := (1 - nn[p]/p)/(1 - 1/p)
bh[n_] := Product[rat[Prime[k]], {k,1,n}]/3

```

