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On the representation of primes by cubic polynomials in two variables

Published online by Cambridge University Press:  08 March 2004

D. R. Heath-Brown  and
B. Z. Moroz
D. R. Heath-Brown
Affiliation:
Mathematical Institute, 24–29 St Giles', Oxford OX1 3LB. E-mail rhb@maths.ox.ac.uk
B. Z. Moroz
Affiliation:
Max-Planck-Institüt für Mathematik, Vivatgasse 7, D-53111 Bonn, Germany. E-mail: moroz@mpim-bonn.mpg.de
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Abstract

In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002) 257–288) we showed that an irreducible integral binary cubic form $f(x, y)$ attains infinitely many prime values, providing that it has no fixed prime divisor. We now extend this result by showing that $f(m, n)$ still attains infinitely many prime values if $m$ and $n$ are restricted by arbitrary congruence conditions, providing that there is still no fixed prime divisor.

Two immediate consequences for the solvability of diagonal cubic Diophantine equations are given.

Keywords

primesbinary cubic polynomialscubic fields11N3211N3611R44
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 88 , Issue 2 , March 2004 , pp. 289 - 312
Copyright
2004 London Mathematical Society

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