TITLE: Primes represented by $x^3-21xy^2+35y^3$.
QUESTION [8 upvotes]: What do we know about the primes represented by this special binary cubic form $x^3-21xy^2+35y^3$?
I know that my question is very short, but I don't have any idea about it, and I don't know where can I find the answer in the literature.
I searched through the net to find a program to check whether a binary cubic equation $f(x, y)=n$ has a solution or not, but I didn't find anything. In the case of the absence of an answer, or a reference to my question, introducing any program/ engine would be welcome.

Let $\alpha$ be a root of the polynomial $x^3-21x-35=0$, and let $K:=\mathbb{Q}(\alpha)$. Then it is easy to show that $$Norm(x+y\alpha+z\alpha^2)=x^3+35y^3+1225z^3-105xyz-21xy^2+441xz^2+42x^2z-735yz^2.$$ That Binary cubic form is just $Norm(x+y\alpha)$.

Note that the discriminant of $P(x)=x^3-21x-35$ is
$-(4\times(-21)^3+27\times(-35)^2)=3969=3^4\times7^2$, so discriminant of $K$ is a square, so it is a cyclic cubic Galois extension, so we can conclude that $r_1=3$ and $r_2=0$. By Dirichlet's unit theorem, we can conclude that $\mathcal{O}_K^{\times}=\{\pm1\}\times\mathbb{Z}^2$. Also, note that $P(x)=x^3-21x-35$ is $7$-Eisenstein, and $P(x-1)=x^3-3x^2+3x-1-21x+21-35=x^3-3x^2-18x-15$ is $3$-Eisenstein; so we can conclude that $\mathcal{O}_K=\mathbb{Z}[\alpha]=\mathbb{Z}\oplus\mathbb{Z}\alpha\oplus\mathbb{Z}\alpha^2$.
Is the answer to the following question positive?

Assume that $Norm(a+b\alpha+c\alpha^2)=p$. Does there exist a unit $u \in \mathcal{O}_K^{\times}$ such that $(a+b\alpha+c\alpha^2)\times u = A+B\alpha$ for some integers $A, B$? Assume that $a+b\alpha+c\alpha^2$ is given. Can we find a suitable unit, such that after the multiplication, we can write the product as linear combination of $1$, and $\alpha$? without need of $\alpha^2$?

REPLY [3 votes]: Let $\alpha$ be a root of $x^3-21x+35=0$. Then it is easy to characterize primes of form $$N(x + y\alpha + z\alpha^2) = x^3+35y^3+1225z^3-105xyz-21xy^2+441xz^2+42x^2z-735yz^2$$
which was already eluded in Will Jagy's answer.

(Theorem) A prime $p\neq 3,7$ can be represented by above cubic form iff $p\equiv \pm 1, \pm 8 \pmod{63}$.

Proof of Theorem: Let $K$ be number field of $x^3-21x+35$. I assume the following facts: $K$ has class number $3$, contained in $\mathbb{Q}(\zeta_{63})$.
Let $H$ be the Hilbert class field of $K$, then $H/\mathbb{Q}$ is abelian of degree $9$ ($H/\mathbb{Q}$ is Galois and any group of order $9$ is abelian).

Claim: $H\subset \mathbb{Q}(\zeta_{63})$. This follows from a general (but not well-known) fact of cyclotomic fields. We have the following proposition, proved in the answer here: If $F/\mathbb{Q}(\zeta_m)$ is unramified (at finite primes) and $F/\mathbb{Q}$ abelian, then $F=\mathbb{Q}(\zeta_m)$. Because $H/\mathbb{Q}$ is abelian, applying this proposition to $F=H\mathbb{Q}(\zeta_{63})$ shows that $H\mathbb{Q}(\zeta_{63}) = \mathbb{Q}(\zeta_{63})$, so $H\subset \mathbb{Q}(\zeta_{63})$.
Claim: $H$ corresponds to $\{\pm 1,\pm 8\} \subset (\mathbb{Z}/63\mathbb{Z})^\times$. $H$ corresponds to a subgroup of order $4$ of $(\mathbb{Z}/63\mathbb{Z})^\times = C_6 \times C_6$, such a subgroup is unique, and this is the only one.

Finally $p\neq 3,7$ can be represented as $N(x + y\alpha + z\alpha^2)$ iff $p$ splits into principle ideals in $K$, iff $p$ splits completely in $H$, completing the proof.

Restricting to $z=0$ of the cubic form is more tricky, and likely has no simple answer. If $\pi(n)$ denotes the prime counting function, then




$p$
No. of $p \equiv 1, 8, 55, 62 \pmod{63}$
No. of $p=x^3-21xy^2+35y^3$




$\pi(p)\leq 3000$
326
61


$3001\leq \pi(p)\leq 6000$
344
42


$6001\leq \pi(p)\leq 9000$
326
32




The equation of form $N(x+y\alpha)$ is a Thue equation. For each individual $p$, there is an algorithm to check whether $N(x+y\alpha) = p$ has integral solution. The following Magma code checks above table for small $p$:
R<x> := PolynomialRing(Integers());
f := x^3 -21*x+35;
T := Thue(f);
list := {71, 127, 181, 197, 251, 307, 379, 433, 449, 503, 631, 701, 757, 811};
t := { n : n in list | Solutions(T, n) ne [] };
t

which outputs { 71, 127, 197, 307, 379, 449, 757 }. The complete list of primes $p$ with $\pi(p)\leq 9000$ which can be written as $p=x^3-21xy^2+35y^3$ is
{71,127,197,307,379,449,757,827,1259,1511,1637,1693,1889,2017,2339,2393,3221,3851,4283,4591,4789,5417,5419,5923,6047,6229,6553,6679,6733,7127,7253,7309,7687,7993,8387,8819,9883,10151,11593,11717,11719,12781,13033,14057,14923,15121,15749,16057,16829,17891,19081,19853,20593,21617,21673,22877,23633,24373,24697,24877,26641,28351,28547,28909,29287,30241,30493,31193,32381,32507,34469,35279,35281,35603,37799,37997,38611,38737,39439,40123,41887,42013,42407,44281,44729,45863,46187,47431,47881,49391,51659,51913,52289,53171,53857,54181,54559,55061,55763,55817,57457,57709,58897,60103,61487,62047,62189,62819,66403,67481,68041,70309,72269,72577,72883,77813,78569,79813,81017,81019,81703,82727,83719,84239,84869,86491,87443,87697,89767,90019,90271,92177,92357,92413,92861}