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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
Affiliation:
Mathematical Institute, 24–29 St Giles', Oxford OX1 3LB. E-mail [email protected]
B. Z. Moroz
Affiliation:
Max-Planck-Institüt für Mathematik, Vivatgasse 7, D-53111 Bonn, Germany. E-mail: [email protected]
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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.

Type
Research Article
Copyright
2004 London Mathematical Society

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