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Primes represented by binary cubic forms. (English) Zbl 1030.11046
Let \(f(x,y)\) be a binary cubic form with integral rational coefficients, and suppose that the polynomial \(f(x,y)\) is irreducible in \(\mathbb Q[x,y ]\) and no prime divides all the coefficients of \(f\). We prove that the set \(f (\mathbb Z^2)\) contains infinitely many primes unless \(f(a,b)\) is even for each \((a,b)\) in \(\mathbb Z^2\), in which case the set \(\frac{1}{2} f(\mathbb Z^2)\) contains infinitely many primes. This theorem is a generalization of the recent theorem of D. R. Heath-Brown [Acta Math. 186, 1–84 (2001; Zbl 1007.11055)] on the infinitude of the primes represented by the cubic form \(x^3+ 2y^3\), and its proof follows the pattern set up in that paper. The main innovations are related to the arithmetic of a general cubic field; our considerations require, in particular, some techniques from E. Hecke’s multidimensional arithmetic which we briefly review. As in the cited work of Heath-Brown, we have actually obtained an asymptotic formula for the relevant number of primes.
Reviewer: B.Z.Moroz (Bonn)

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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