When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalous subvarieties. These arise when the algebraic variety meets translates of subgroups in sets larger than expected. We prove a Structure Theorem for the anomalous… (More)
1 Voltaire said that Archimedes had more imagination than Homer. Unfortunately, as far as we know, Archimedes's only use of it, outside of mathematics, was to condemn goldsmiths and to kill people. More peaceful uses of mathematicians' imagination, to the outside of mathematics, in decreasing order of impact, were provided by Multi-Millionaire Richard… (More)
The study of the distribution of prime numbers has fascinated mathematicians since antiquity. It is only in modern times, however, that a precise asymptotic law for the number of primes in arbitrarily long intervals has been obtained. For a real number x > 1, let π(x) denote the number of primes less than x. The prime number theorem is the assertion that… (More)
I. Let Q(N; q; a) denote the number of squares in the arithmetic progression qn+a; n = 1; 2; ; N; and let Q(N) be the maximum of Q(N; q; a) over all non-trivial arithmetic progressions qn + a. It seems to be remarkably diicult to obtain non-trivial upper bounds for Q(N). There are currently two proofs known of the weak bound Q(N) = o(N) (which is an old… (More)
Our investigations in the 1980s of Thue's method yielded determinants which we were only able to analyse successfully in part. We explain the context of our work, recount our experiences, mention our conjectures, and allude to a number of open questions.
This paper seeks to explain in the simplest terms possible a paper written by Umberto Zannier. Though Zannier says that his is " a simple elementary method, " there are still steps in his paper that are quite subtle. The tools needed to follow his proof are in the hands of most Algebra students, though which tools to use and how to use them may not be… (More)
Hilbert delivered his famous lecture in which he described twenty-three problems that were to play an infl uential role in mathematical research. A century later, on May 24, 2000, at a meeting at the Collège de France, the Clay Mathematics Institute (CMI) announced the creation of a US$7 million prize fund for the solution of seven important classic… (More)
The Riemann zeta function is the function of the complex variable s, defined in the half-plane 1 (s) > 1 by the absolutely convergent series ζ(s) := ∞ n=1 1 n s , and in the whole complex plane C by analytic continuation. As shown by Riemann, ζ(s) extends to C as a meromorphic function with only a simple pole at s = 1, with residue 1, and satisfies the… (More)