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On the number of solutions of polynomial congruences and Thue equations
has only finitely many solutions in integers x and yv. In the first part of this paper we shall establish upper bounds for the number of solutions of (1) in coprime integers x and y under the
On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms
Let E be an elliptic curve over Q. By the rank of E we shall mean the rank of the group of rational points of E. Mestre [31], improving on the work of Neron [34] (cf. [13], [39] and [46]), has shown
On the representation of an integer in two different bases.
In 1970 Senge and Strauss [4] proved that the number of integers, the sum of whose digits in each of the bases a and b lies below a fixed bound, is fmite if and only log if is irrational. Their
On the Oesterlé-Masser conjecture
Letx, y andz be positive integers such thatx=y+z and ged (x,y,z)=1. We give upper and lower bounds forx in terms of the greatest squarefree divisor ofx y z.
On the number of prime factors of integers of the form ab + 1
If the sets A and B are dense sets of integers then estimates (1.1) and (1.2) may be strengthened. Let and δ be positive real numbers and let N The research of the first two authors was partially
A refinement of the abc conjecture
Based on recent work, by the first and third authors, on the distribution of the squarefree kernel of an integer, we present precise refinements of the famous abc conjecture. These rest on the sole
On divisors of sums of integers. II.
Throughout this article, c0, cl9 c2,... will denote effectively computable positive absolute constants. Denote the cardinality of a set X by \X\ and for any integer n let P (n) denote the greatest