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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
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