Let $ax^{2}+bxy+cy^{2}$ be a primitive positive definite quadratic form of discriminant $\Delta<0$. It is well known that $ax^{2}+bxy+cy^{2}$ represents infinitely many prime numbers. One of the proofs of this result is elementary and the other, which is more oldest, is based on the Chebotarev density theorem.

My question is if there is other similar results, based on the Chebotarev density theorem, and implying the existence of forms (with degree more than $2$, more variables than $2$) representing infinitely many prime numbers (or other family of numbers).

I would know if there is any other significant results like the theorem above, where the Chebotarev density theorem must be a fundamental ingredient of their proof. I need references.

Thank you for any help.

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    $\begingroup$ Where can we find the elementary proof? $\endgroup$ – GH from MO Jun 22 '14 at 7:53
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    $\begingroup$ I would recommend reading Heath-Brown's article "Primes Represented by $x^3 + 2y^3$" if you have not do so already. I have not gone through enough of the 76 pages to see whether he uses the Chebotarev density theorem, but I would be surprised if he did not. When there are many variables one usually uses the circle method, rather than the Chebotarev density theorem, to show that a homogeneous polynomial represents infinitely many primes. $\endgroup$ – Daniel Loughran Jun 22 '14 at 9:37
  • $\begingroup$ @GHfromMO, see mathoverflow.net/questions/144544/… where the 1954 proof by Briggs really is elementary, uses results of Selberg developed for the Prime Number Theorem $\endgroup$ – Will Jagy Jun 22 '14 at 16:24
  • $\begingroup$ Related: mathoverflow.net/questions/28280. The proof using Chebotarev is in Cox's book, theorem 9.12. $\endgroup$ – Watson Dec 3 '18 at 10:45
  • $\begingroup$ See also "NORM FORMS REPRESENT FEW INTEGERS BUT RELATIVELY MANY PRIMES" by DANIEL GLASSCOCK. $\endgroup$ – Watson Dec 3 '18 at 10:47

The example of Heath-Brown's article is a good one. For a bit more elementary examples, you can fix a number field $K$ with $[K : \mathbb{Q}] = n$ and ring of integers $\mathcal{O}_{K}$ and pick a basis $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$. (I will assume that $n$ is odd for convenience - this guarantees the existence of a unit of norm $-1$.) Then, let $$ P(x_{1}, x_{2}, \ldots, x_{n}) = N_{K/\mathbb{Q}}\left(\sum_{i=1}^{n} x_{i} \alpha_{i}\right). $$ The polynomial $P$ will have integer coefficients and be a degree $n$ form in $n$ variables. The polynomial $P$ will represent a prime number $p$ if and only if there is a principal prime ideal in $\mathcal{O}_{K}$ with norm $p$. The existence of infinitely many such ideals follows from the Chebotarev density theorem. (It seems likely to me a modification of this construction can obtain forms corresponding to non-trivial classes in the ideal class group, but I don't see how right now.)

For example, if $K = \mathbb{Q}[\sqrt[3]{-7}]$, then we can take $$ P(x,y,z) = x^{3} + 7y^{3} + 49z^{3} - 21xyz.$$ The ring of integers in $K$ is $\mathcal{O}_{K} = \mathbb{Z}[\sqrt[3]{-7}]$ has class number $3$, and $P$ represents the primes $7$, $29$, $41$, $71$, $83$, $\ldots$ that split in the Hilbert class field of $K$. The number of such primes $\leq x$ is asymptotic to $\frac{2}{9} \frac{x}{\log(x)}$.

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    $\begingroup$ To get nontrivial classes in the class group, start with an ideal $A$ representing the inverse of the class you want. Elements of $A$ modulo units are in bijection with ideals of class $A^{-1}$ (the bijection is $\alpha \mapsto \alpha A^{-1}$). Choose an integer basis $(x_1, \ldots, x_n)$ of $A$. Then prime ideals of class $A^{-1}$ correspond to prime values of the polynomial $N(\sum a_i x_i)/N(A)$. $\endgroup$ – DES-SupportsMonicaAndTransfolk Jun 22 '14 at 12:10
  • $\begingroup$ Jeremy, see mathoverflow.net/questions/144544/… The 1954 article of Briggs is ``elementary'' $\endgroup$ – Will Jagy Jun 22 '14 at 16:25
  • $\begingroup$ Jeremy, please also see mathoverflow.net/questions/172432/… $\endgroup$ – Will Jagy Jun 22 '14 at 17:18

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