Proof (Continued from the previous class) We have already shown in the previous class that the algorithm halts after polynomially many executions of the while-loop as the index i can decrease for at most O(n logA) times in step 6. In order to complete the proof, we also need to show that the size of the numerator and denominator of any rational number involved in the computation is polynomially...

We introduce the notion of wild partition to describe in combinatorial language an important situation in the theory of p-adic fields. For Q a power of p, we get a sequence of numbers λQ,n counting the number of certain wild partitions of n. We give an explicit formula for the corresponding generating function ΛQ(x) = ∑ λQ,nx and use it to show that λ Q,n tends to Q 1/(p−1). We apply this asymp...

“Combinatorial number theory”, in very loose terms, can be described as an area of mathematics which is a cross between combinatorics and number theory. More precisely, the area concerns structures of integers (or similar sets), with some number theoretic properties, which can be studied mainly by combinatorial means, rather than, for example, by algebraic methods. This area really only truly e...

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Ea...

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of a...

An excellent introduction to the theory of automorphic representations and the relations with number theory is the Corvallis proceedings, Automorphic Forms, Representations and L-Functions, Parts 1 and 2, Proc. Sympos. Pure Math., vol. 33, 1979. Although composed mainly of survey articles, the proceedings are already rather formidable. They are a measure of the breadth of the field. They will b...