On the Representation of Primes by Cubic Polynomials in Two Variables

In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002)257–288) we showed that an irreducible integral binarycubic form f(x, y) attains infinitely many prime values, providingthat it has no fixed prime divisor. We now extend this resultby showing that f(m, n) still attains infinitely many primevalues 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 cubicDiophantine equations are given. 2000 Mathematics Subject Classification11N32 (primary), 11N36, 11R44 (secondary).     

小区间中的整数的最大素因子

设ε>0,证明了:区间(X,X十X^1/2十ε]中存在这样一个整数,它有一个素因子P≥x^11/12-ε.     

The Density of Rational Points on Non-Singular Hypersurfaces, I

For any n 3, let F Z[X₀, ..., X_n] be a form of degree d 5that defines a non-singular hypersurface X Pⁿ. The main resultin this paper is a proof of the fact that the number N(F; B)of Q-rational points on X which have height at most B satisfies , for any > 0. The implied constantin this estimate depends at most upon d, and n. New estimatesare also obtained for the number of representations of a positiveinteger as the sum of three dth powers, and for the paucityof integer solutions to equal sums of like polynomials. 2000Mathematics Subject Classification 11G35 (primary), 11P05, 14G05(secondary).     

Counting rational points on cubic curves

We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a combination of the “determinant method” with an m-descent on the curve.     

Bounds for the cubic Weyl sum

Subject to the abc-conjecture, we improve the standard Weyl estimate for cubic exponential sums in which the argument is a quadratic irrational. Specifically. we show that