
Abstracts



Jonathan Borwein, Dalhousie University
"Ramanujan's AGM continued fractions and their dynamics"
An innnocent computational questionarising with an identity due to
Ramanujan led me, Richard Crandall, and our other collaborators into
an exhilerating empirical ride through a rich garden of continued
fractions, elliptic integrals, special functions and dynamical
systems. I invite you to take a guided tour through this garden. It
has now been tamed in that all is proven, as I shall indicate.
David Bradley, University of Maine
"On qanalogs of multiple zetavalues and other multiple harmonic series"
An example of a multiple harmonic sum is
$Z_n(s,t,u):= \sum_{n\ge i\ge j\ge k} i^{s} j^{t} k^{u}$.
Here, the sum is over all positive integers $i$, $j$, $k$
satisfying the indicated inequalities, which in some cases
may be strict instead of weak as indicated. The bound $n$
may be finite or infinite. The arguments
$s$, $t$, $u$ are unrestricted if $n$ is finite, but are
usually assumed to be positive integers, with $s>1$ if
$n$ is infinite to ensure convergence. In general, we
may have an arbitrary finite number of arguments, as
opposed to only three.
The $q$analog of a positive integer $k$ is
$[k]_q:= \sum_{j=0}^{k1}q^j = (1q^k)/(1q)$, $q\ne 1$.
Naively, one can obtain a $q$analog of multiple harmonic
sums by replacing the summation indices by their respective
$q$analogs. However, it appears that this approach needs
to be modified somewhat in order to yield interesting results.
I plan to discuss what sorts of results can be obtained with
appropriate modifications, outline a few of the techniques
used to prove them, and hopefully give an indication of why
researchers are interested in this subject.
Marc Chamberland, Grinnell College
"An update on the 3x+1 problem"
The 3x+1 Problem is a longstanding conjecture in number
theory. Let T be a map from the positive integers into
itself, where T(x)=x/2 if x is even and T(x) = (3x+1)/2
if x is odd. The conjecture asks whether, under iteration
of the map T, any positive integer eventually reaches the
value one. This talk gives a survey of the
various approaches and results.
Robin Chapman, University of Exeter
"Determinants of Legendre symbol matrices"
We consider Hankel and Toeplitz matrices built from Legendre
symbols. In patricular we consider their determinants. Computation
suggests various conjectures concerning their evaluation; only some
of which we have proved.
Stephen Choi, Simon Fraser University
"Linear Diophantine Equation with Three Prime Variables"
In this talk, we will discuss the small prime solutions of the
following linear diophantine equation with three prime variables:
a_1 p_1 + a_2 p_2 + a_3 p_3 = b.
Mark Coffey, Colorado School of Mines
"New results on the Stieltjes constant: Proof of a Kreminski conjecture"
The Stieltjes constants [1] \zeta_k have been of interest for over a century,
yet their detailed behavior remains under investigation.
We have derived new relations amongst the values \zeta_k(a) that are the
expansion coefficients in the Laurent series for the Hurwitz zeta function.
In this talk, we demonstrate the validity of one of the very recent
conjectures of Kreminski [2] on the relationship between \zeta_k(a)
and \zeta_k(a+1/2) as k > \infty. In addition, we have developed a new
integral relationship for the Hurwitz zeta function and one of
its implications for the Stieltjes constants.
[1] T. J. Stieltjes, Correspondance d'Hermite et de Stieltjes,
Volumes 1 and 2, GauthierVillars, Paris (1905).
[2] R. Kreminski, NewtonCotes integration for approximating Stieltjes
(generalized Euler) constants, Math. Comp. 72, 13791397 (2003).
Eva Curry, Rutgers University and Dalhousie
"Radix representations of Z^n"
An integer b gives a radix representation if there is a set of digits
D such that every integer x can be written in the form
$J = \sum_{j=0}^{N(x)} b^j d_j$,
with the $d_j \in D$. This talk will present a generalization
of this idea to Z^n. I will give a sufficient
condition for a matrix A to give a radix representation for
Z^n, and will discuss a generalization of signed radix
representations. I will also show how the question of which matrices give
a radix representation for Z^n is related to selfaffine
tilings of R^n, and discuss connections to wavelets.
Karl Dilcher, Dalhousie University
"A Pascaltype triangle characterizing twin primes"
It is a wellknown property of Pascal's triangle that the entries
of the kth row, without the initial and final entries 1, are all
divisible by k if and only if k is prime. In this talk I will
present a triangular array similar to Pascal's that characterizes
twin prime pairs in a similar fashion. The proof involves generating
function techniques. Connections with orthogonal polynomials, in
particular Chebyshev and ultraspherical polynomials, will also be
discussed. This is joint work with K.B. Stolarsky.
Kevin Hare, University of Waterloo
"Odd perfect numbers"
A perfect number is a number $n$, such that the some of all
perfect divisors of $n$ sum to $n$. The first two examples are 6 and
28. All known examples of perfect numbers are even. The existance
of odd perfect numbers is still undecided, and is the area of active
research. This talk will discuss some of the history of the search for
odd perfect numbers, as well as some of the computation techniques
used to find lower bounds on the existance or nonexistance of odd
perfect numbers.
Jeff Hooper, Acadia University
"Explicit Galois Module structure for Quaternion Extensions of Local Fields"
Let $K$ be finite extension of $\bQ_2$, the $2$adic numbers, and let
$N/K$ be a quaternion extension of degree $8$. Although these extensions
have been fairly wellstudied over the years, most of the work on these has been nonexplicit.
In a current project with Griff Elder, we look to explicitly describe the
$\bZ_2[Gal(N/K)]$module structure of each fractional ideal of $N$ in
terms of indecomposables. In this talk I will report on some recent
progress in this direction.
Keith Johnson, Dalhousie University
"Formal group laws and Legendre numbers"
Lutz Lucht, Technische Universität Clausthal
"Mean behaviour of arithmetic functions on certain thin sequences"
Meanvalue theorems with quantitative remainder term estimates are proved
for large classes of arithmetic functions $f$ on certain thin sequences
$\mathfrak{a}$ of positive integers. These classes depend on the
distribution of the divisors of $\mathfrak{a}$ and on convergence
properties of the Dirichlet series of the quotient
$\widetilde{f}(s)/\zeta(s)$, where $\widetilde{f}(s)$ is the Dirichlet
series associated with $f$, and $\zeta(s)$ denotes the Riemann zeta
function. The results are visualized for specific multiplicative functions
$f$ and sequences $\mathfrak{a}$\,.
Carl Pomerance, Dartmouth College
"New primality criteria, after Agrawal, Kayal, and Saxena"
Two years ago, M. Agrawal, N. Kayal, and N. Saxena announced their
new deterministic, polynomial time primality test. They proved
that their primality test would decide if $n$ is prime or composite
in about (log n)^12 elementary steps, and they conjectured the
true complexity was
about (log n)^6. Since then the exponent 12 has crept down
due to their efforts and others, and it now stands at 6, the original
conjecture, albeit for a somewhat different test. This last is
a joint result of me and Hendrik Lenstra. Towards a more
practical primality test, we even have a random version due to Bernstein,
with expected running time of about (log n)^4.
Although the complexity estimates
are sometimes not easy to establish, these tests are all based on
succinct and essentially elementary primality criteria. These
talks will give an overview of the different primality tests,
and prove some of the criteria.
Sinai Robins
"Hecke operators on rational functions and the Artin conjecture for
primitive roots"
Kenneth B. Stolarsky, University of Illinois at UrbanaChampaign
"Problems about polynomials related to Chebyshev's (discriminants, zeros,
qanalogues, etc.)"
Chebyshev polynomials are defined by a 3term linear recurrence
whose coefficients are (extremely simple) polynomials in x. We know their zero
distribution because (e.g.) they are orthogonal polynomials. What happens if we
move out of the orthogonality realm by changing the coefficient polynomials?
Amazingly little seems to be known here. We indicate how in certain carefully
chosen cases this may have connections with previous DilcherStolarsky results
on resultants of linear combinations of Chebyshev polynomials, and (more
speculatively) how it may connect with the distribution of the powers
3^n modulo p.
Keith Taylor, Dalhousie University
"Asymptotic properties of the spectra of Toeplitz matrices"
Let $T$ be the Toeplitz operator with symbol
$\phi$. Here $\phi$ is a bounded measurable function
on the unit circle and $T$ is an operator on $l^2$ that,
considered as an infinite matrix, has the Fourier coefficient
of $\phi$ at $ij$ as its $i,j$ entry. If the symbol is
a polynomial, then $T$ has only a finite number of nonzero
diagonals (it is "bandlimited"). For any positive integer
$n$, let $T_n$ denote the $n$ by $n$ upper left corner matrix
of $T$. Szego theory gives the limiting distribution of the
spectra of the $T_n$ as $n$ goes to infinity. We present
an estimate on the rate at which the distribution of the
spectra of the $T_n$ approach this limiting distribution.
Alf van der Poorten, Centre for Number Theory Research, Sydney
"Elliptic sequences and continued fractions"
The symmetric sequence
..., 2, 1, 1, 1, 1, 2, 3, 7, 23, 59, ...
defined by
$A_{h2}A_{h+2}=A_{h1}A_{h+1}+A_{h}^2$ and $A_0=A_1=A_2=A_3=1$
arises from the curve $V^2V=U^3+3U^2+2U$ by reporting the
denominators of the points $M+hS$, with $M=(1,1)$ and $S=(0,0)$.
The sequence (..., 2, 1,1,1,1,1, 1, 2,3,4,8,17,50, ... )
is given by the recursion $B_{h3}B_{h+3}=B_{h2}B_{h+2}+B_{h}^2$ and
$B_0=B_1=B_2=B_3=B_4=B_5=1$ and arises from adding multiples of the
divisor at infinity on the Jacobian of the curve
$Y^2=(X^34X+1)^2+4(X2)$ of genus 2 to the divisor given by
$[(\varphi,0),(\conj\varphi,0)]$; here, no doubt to the joy of
adherents to the cult of Fibonacci, $\varphi$ is the golden ratio.
Of course the real surprise is that the stated recursions produce
sequences of integers.
Gary Walsh, University of Ottawa
"On a diophantine equation of Cassels"
J.W.S. Cassels gave a solution to the problem of determining all
instances of the sum of three consecutive cubes being a square.
This amounts to
finding all integer solutions to the Diophantine equation $y^2=3x(x^2+2)$.
We describe an alternative approach to solving not only this equation, but
any equation of the type $y^2=nx(x^2+2)$, with $n$ a natural number.
Moreover, we provide an explicit upper bound for the number of solutions of
such Diophantine equations.
The method we present uses the ingenious work of Wilhelm Ljunggren, and a
recent improvement by Florian Luca and the speaker.
We further provide an upper bound for the number of solutions to
the more general equation $y^p=nx(x^2+2)$, with $p$ prime, by applying
a very recent, and very deep, result of Michael Bennett. This is joint work
with Florian Luca.
