Past Junior Number Theory Seminar

24 October 2011
16:00
Sebastian Pancratz
Abstract
<p>We describe various approaches to the problem of expressing a polynomial $f(x) = \sum_{i=0}^{m} a_i x^i$ in terms of a different radix $r(x)$ as $f(x) = \sum_{j=0}^{n} b_j(x) r(x)^j$ with $0 \leq \deg(b_j) &lt; \deg(r)$. Two approaches, the naive repeated division by $r(x)$ and the divide and conquer strategy, are well known. We also describe an approach based on the use of precomputed Newton inverses, which appears to offer significant practical improvements. A potential application of interest to number theorists is the fibration method for point counting, in current implementations of which the runtime is typically dominated by radix conversions.</p>
  • Junior Number Theory Seminar
17 October 2011
16:00
Jan Vonk
Abstract

The theory of modular forms owes in many ways lots of its results to the existence of the Hecke operators and their nice properties. However, even acting on modular forms of level 1, lots of basic questions remain unresolved. We will describe and prove some known properties of the Hecke operators, and state Maeda's conjecture. This conjecture, if true, has many deep consequences in the theory. In particular, we will indicate how it implies the nonvanishing of certain L-functions.

  • Junior Number Theory Seminar
10 October 2011
16:00
James Maynard
Abstract

We discuss conjectures and results concerning small gaps between primes. In particular, we consider the work of Goldston, Pintz and Yildrim which shows that infinitely often there are gaps which have size an arbitrarily small proportion of the average gap.

  • Junior Number Theory Seminar
21 June 2011
14:00
Jan Tuitman
Abstract
(Note change in time and location) The purpose of this talk is to give an introduction to the theory and practice of integer factorization. More precisely, I plan to talk about the p-1 method, the elliptic curve method, the quadratic sieve, and if time permits the number field sieve.
  • Junior Number Theory Seminar
9 May 2011
16:00
163
Frank Gounelas
Abstract
<p>I will describe why e^{\pi\sqrt{163}} is almost an integer and how this is related to Q(\sqrt{-163}) having class number one and why n^2-n+41 is prime for n=0,...,39. Bits and pieces about Gauss's class number problem, Heegner numbers, the j-invariant and complex multiplication on elliptic curves will be discussed along the way.</p>
  • Junior Number Theory Seminar

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