×

zbMATH — the first resource for mathematics

On the representation of primes by cubic polynomials in two variables. (English) Zbl 1099.11050
Let \(f_0(x, y) \in {\mathbb Z}[x, y]\) be a binary cubic form that is irreducible in \({\mathbb Z}[x, y]\). In a recent paper [Proc. Lond. Math. Soc. (3), 84, No. 2, 257–288 (2002; Zbl 1030.11046)], the authors proved that there exist infinitely many primes of the form \(f_0(a, b)\), \((a, b) \in {\mathbb Z}^2\), unless \(2\) divides \(f_0(a, b)\) for all \((a, b) \in {\mathbb Z}^2\); in the latter case, there exist infinitely many primes of the form \(\frac 12f_0(a, b)\), \((a, b) \in {\mathbb Z}^2\). In the paper under review, they extend that result to a wider class of polynomials. With \(f_0(x, y)\) as before, let \(d, u, v \in {\mathbb Z}\) and let \(c \in {\mathbb Z}\) be chosen so that \(f(x, y) = c^{-1}f_0(u + dx, v + dy)\) is a primitive polynomial in \({\mathbb Z}[x, y]\) (that is, its coefficients have no common divisor other than \(\pm 1\)). The main theorem of the paper establishes an asymptotic formula for the number of primes of the form \(f(a, b)\), \((a, b) \in {\mathbb Z}^2\).
Several applications to rational points on cubic surfaces are also included. For example, one corollary states that if \(a\) and \(b\) are coprime integers such that \(a \equiv \pm b \pmod 9\), then the surface \(x_0^3 + 2x_1^3 + ax_2^3 + bx_3^3 = 0\) contains a nontrivial rational point. The authors remark that such a result cannot be deduced from their earlier work on binary cubic forms.

MSC:
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11N36 Applications of sieve methods
11R44 Distribution of prime ideals
PDF BibTeX Cite
Full Text: DOI