Number Theory at UBC

SFU/UBC Number Theory Seminar
Fall Semester 2002


Thursday, September 12, 2002
PIMS Seminar Room (West Mall Annex 216), UBC Campus

3:00-3:45
Colin Graham (University of British Columbia)
Which subsets of the roots of unity are Sidon in the plane?

Abstract: The motivating question is this: how does the multiplicative structure of the roots of unity relate to the additive structure of the complex plane? More specifically, what subsets of the roots of unity are Sidon sets (that is, have no repeated pairwise sums) in the discrete plane? The approach is essentially combinatorial, and no knowledge of harmonic analysis will be needed.

For any finite set of positive primes, P, let W be the be multiplicative subset of Z generated by P.  Then
E={e2πia/m | a∈Z and m∈W}
is a Sidon subset of the additive group of complex numbers. In fact, E is a finite union of M independent sets, where
M=⌈Πq∈P q/(q-1)⌉.
More generally, a subset S⊂eπiQ is Sidon if and only if its intersections with cosets of certain (multiplicative) subgroups, namely those with square-free order, are Sidon uniformly.

We characterize the permutations of the roots of unity that preserve the quasi-independent sets (sets whose subsets all have distinct sums); they are the same permutations that preserve the independent sets. (joint work with L. T. Ramsey)
3:45-4:15
tea break
4:15-5:00
Greg Martin (University of British Columbia)
The Agrawal-Kayal-Saxena deterministic polynomial-time primality test

Abstract: How quickly can we determine whether a given (enormous) number is prime?  We would like to have a polynomial-time algorithm—that is, a systematic procedure that will never take more than P(d) arithmetical operations to complete, where P(x) is some polynomial and d is the number of digits in our enormous number—for testing primality.  Until last month(!), our best primality tests used randomization and therefore could in principle give the wrong answer, or else could only be proven to run in polynomial-time if we assumed the Extended Riemann Hypothesis.  But the three Indian computer scientists in the title have recently announced a test for primality, not involving randomization, that can be proven to run in polynomial time, without having to assume anything like the ERH.  In this talk, which will be accessible to graduate students and nonspecialists, I will describe the previously known primality tests and then present the new AKS test.


Thursday, September 26, 2002
PIMS Seminar Room (East Academic Annex Room 1100), SFU Campus
3:00-3:45
Will Galway (PIMS/SFU)
BBP formulae of Machin-type

Abstract: Constants with formulae of the form treated by D. Bailey, P. Borwein, and S. Plouffe (BBP formulae to a given base b) have interesting computational properties, and there are hints that they should be “normal” numbers, i.e., that their base-b digits are randomly distributed.  We study a formally limited subset of BBP formulae, which we call Machin-type formulae, for which it relatively easy to determine whether or not a given constant C has a Machin-type formula.  In particular, given an integer b > 2 that is not a proper power, a b-ary Machin-type arctangent formula for C is a formula of the form
C = Σ am arctan(-b-m),
where the sum is over positive integers m and where each amQ. (When b = 2 we also allow terms of the form am arctan(1/(1-2m )).) Of particular interest, we show that π has no Machin-type arctangent formula when b ≠ 2.  To the best of our knowledge, when there is no Machin-type formula for a constant then no BBP formula of any form is known for that constant. (joint work with J. Borwein and D. Borwein)
3:45-4:15
tea break
4:15-5:00
Michael Bennett (University of British Columbia)
Products of consecutive integers

Abstract: A celebrated theorem of Erdős and Selfridge states that the product of consecutive positive integers can never be a perfect power. In this talk, I will discuss some recent work generalizing this result, from a variety of perspectives.


Thursday, October 10, 2002
PIMS Seminar Room (West Mall Annex 216), UBC Campus
3:00-3:45
Alexa van der Waall (MITACS/SFU)
Zeros of truncated zeta-functions and the Riemann hypothesis

Abstract: In 1948 Paul Turán proved several theorems relating the Riemann Hypothesis to the behaviour of the zeros of the partial sums
ζN(z) = Σ1≤n≤N 1/nz    (N>1)
of the Riemann zeta-function, called “truncated zeta-functions”.  The behaviour of the zeros of the “alternating truncated zeta-functions”
Σ1≤n≤N (-1)n+1/nz    (N>1)
have a similar importance.  We discuss some of Turán's statements and some more resent results.  In particular, we present several pictures of zeros of the truncated zeta-functions that result in specific curves for small values of N.
3:45-4:15
tea break
4:15-5:00
Alina Cojocaru (University of Illinois/Fields Institute)
Elliptic curves modulo p

Abstract: Let E be an elliptic curve defined over the rationals. For a prime p, let Ep be the reduction of E modulo p. We will discuss a few questions concerning the group of points of Ep as p varies, such as how often we have cyclicity or square-free order for this group. These questions arise naturally when one considers elliptic curve analogues of classical problems in number theory: Artin's primitive root conjecture, Linnik's problem, twin prime conjecture.


Thursday, October 24, 2002
PIMS Seminar Room (East Academic Annex Room 1100), SFU Campus
3:00-3:45
Idris Mercer (Simon Fraser University)
Moments of autocorrelations of Littlewood polynomials

Abstract: A polynomial all of whose coefficients are either +1 or -1 is called a Littlewood polynomial, and we can identify such a polynomial with its sequence of coefficients.  The autocorrelations of a sequence of +1's and -1's are essentially dot products which measure the correlation between that sequence and “shifted” versions of itself.  A sequence with autocorrelations “near zero” tends to correspond to a polynomial that is “flat” on the unit circle.  Some information about the “flatness” of the polynomial or the “smallness” of its autocorrelations can be obtained by supposing the coefficients are independent random variables, each equally likely to be +1 or -1.  For example, we can find the expected value of the rth power of the autocorrelations for any positive integer r.
3:45-4:15
tea break
4:15-5:00
Hugh Edgar (San Jose State University, emeritus)
Sum problems in number theory

Abstract: As a warm-up for knocking off Hilbert's 10th problem, Matiyasevich proved that
if Fn2 divides Fm, then Fn divides m,
where Fk is the kth Fibonacci number.  We will present a proof of this (one more rigorous than simply dividing both sides by F) and other interesting tidbits.


Thursday, November 7, 2002
UBC Campus, PIMS Seminar Room (West Mall Annex 216)
3:00-3:45
Peter Borwein (Simon Fraser University)
The Merit Factor problem of Golay

Abstract: The Merit Factor problem is an old and very difficult combinatorial optimization problem.  It seeks to maximize the Merit Factor of a binary sequence (or equivalently minimize the L4 norm on the unit disc of a polynomial with ±1 coefficients).  The best known examples are constructed out of rotated sequences of Legendre symbols and are quite surprising.
3:45-4:15
tea break
4:15-5:00
Imin Chen (Simon Fraser University)
On Drinfeld modular curves

Abstract: I will discuss address some of the difficulties in applying Mazur's method to study the rational points on Drinfeld modular curves.


Thursday, November 21, 2002
SFU Campus, PIMS Seminar Room (East Academic Annex Room 1100)
3:00-3:45
Stephen Choi (Simon Fraser University)
Small prime solutions of quadratic equations

Abstract: Let b1, …, b5 be non-zero integers and n any integer.  Suppose that b1 + … + b5 ≡ n (mod 24) and (bi,bj) = 1 for 1 ≤ i < j ≤ 5.  In this talk, we will discuss two new results: (i) if the bj are not all of the same sign, then the quadratic equation
b1p12 + … + b5p52 = n
has prime solutions satisfying pj « √n + max{|bj|}25/2+ε; and (ii) if the bj are all positive and n » max{|bj|}26+ε, then the above quadratic equation is soluble in primes pj.  Our previous results are max{|bj|}20+ε and max{|bj|}41+ε in place of max{|bj|}25/2+ε and max{|bj|}26+ε above, respectively. This is joint work with Jianya Liu.
3:45-4:15
tea break
4:15-5:00
David Boyd (University of British Columbia)
Random explanations of non-random phenomena in elementary number theory

Abstract: Simple probabalistic arguments often give suprisingly accurate predictions regarding questions about the integers.  We will present a number of examples to illustrate this claim.  The general question of why this method is so successful will be left to the audience to ponder.


Thursday, December 5, 2002
UBC Campus, PIMS Seminar Room (West Mall Annex 216)
3:00-3:45
Michael Bennett (University of British Columbia)
Catalan's Conjecture I
3:45-4:15
tea break
4:15-5:00
Ron Ferguson (MITACS/SFU/UBC)
Catalan's Conjecture II
Abstract: An old problem, dating back at least to Eugene Catalan in the 19th century, is whether 8 and 9 are the only two consecutive perfect powers (of positive integers).  Despite the simplicity of its statement, this problem resisted the efforts of some of the world's greatest mathematicians until 1976, when Rob Tijdeman showed that if xp - yq = 1 (where x, y, p and q are positive integers with p and q at least 2), then x, y, p and q are bounded by an effectively computable constant.  This reduces the problem to a finite (though extraordinarily large) computation.

In the last year, Preda Mihailescu surprised everyone with a strikingly elegant complete proof that 8 and 9 are in fact the only consecutive powers.  His proof is very different than that of Tijdeman, depending very little on techniques from transcendental number theory.  In these two talks, we will begin by surveying the (vast) literature on Catalan's conjecture and sketch proofs of a number of partial results due to Cassels and Chao Ka, before proceeding with that of Tijdeman.  In the second part, we will outline Mihailescu's insights and show how they write the final chapter in a remarkable tale.