You want primes
$$ p = 7 x^2 + 6 xy + 36 y^2 $$
with integers $x,y.$ Notice that such primes are represented by $4 s^2 + 2 st + 7 t^2$ and $7 u^2 + 3 uv + 9 v^2,$ so both $2$ and $3$ are not cubic residues. Here one could just write $s = 3y, t = x,$ for the other $u=x, v = 2y.$

Meanwhile, in $ p = 7 x^2 + 6 xy + 36 y^2 $ we could take $x=1$ and $y = 6k$ to arrive at your $7 + 36 k + 1296 k^2$

```
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primego
Input three coefficients a b c for positive f(x,y)= a x^2 + b x y + c y^2
7 6 36
Discriminant -972
Modulus for arithmetic progressions?
3
Maximum number represented?
3000
7, 37, 139, 163, 181, 241, 313, 337, 349, 379,
409, 421, 541, 571, 607, 631, 751, 859, 877, 937,
1033, 1087, 1123, 1171, 1291, 1297, 1447, 1453, 1483, 1693,
1741, 1747, 2011, 2161, 2239, 2311, 2371, 2473, 2539, 2647,
2677, 2707, 2719, 2857,
```

Bunyakowsky's Conjecture. You will find there isnoone-variable quadratic for which it has been proved that the quadratic takes on infinitely many prime values. $\endgroup$ – Gerry Myerson Mar 26 at 8:55