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The generalized superelliptic equation
by
Michael Bennett
University of British Columbia
Coauthors: Sander Dahmen
Given an irreducible binary form F(x, y) of degree at least three, one might expect that the form represents at most finitely many kth powers, for k > 3 variable. While such a conclusion does follow from a suitable number field generalization of the ABC-conjecture, it has not been previously possible to exhibit, for example, a single cubic form for which we can prove it to hold. In this talk, I will sketch recent joint work with Sander Dahmen which establishes this result for a large class of forms. We rely upon new ideas from the theory of Frey curves and their associated Galois representations.
Date received: October 29, 2008
Rational solutions of y2=x6+k
by
Andrew Bremner
Arizona State University
Coauthors: Nikos Tzanakis
We discuss techniques for finding all rational points on the genus 2 curves y2=x6+k, and apply these techniques to find all solutions of the title equation for |k| ≤ 50, with the exception of k=-47, -39.
Date received: October 16, 2008
On the missing values of n! mod p
by
Kevin Broughan
University of Waikato
Coauthors: A. Ross Barnett
Consider the question of the values of n! mod p for odd primes p and 1 ≤ n ≤ p-1. Numerical evidence shows that about p/e of the residue classes are missing, but there has been little progress in explaining this phenomenon. That is except for a result of Cobeli, Vajaitu and Zharescu [2] who show that the 1/e missing values proportion arises as an average when the set of all sequences is considered.
Here we show how the 1/e proportion arises, and then give some details of the recent result that when sequences are considered which obey a "no identical neighbors" condition, and so better model the factorial, the mysterious 1/e proportion is maintained.
[1] Broughan, K. A. and Barnett, A.R, On the missing values of n! mod p, (submitted 2008).
[2] Cobeli, C., Vâjâitu, M., and Zaharescu, A. The sequence n! ( mod ) p, J. Ramanujan Math. Soc. 15 (2000), p135-154.
Date received: September 7, 2008
Series and iterations for 1/p
by
Shaun Cooper
Massey University
I will show how the Rogers-Ramanujan continued fraction and four other similar functions can be used to derive series and iterations for 1/p. This extends recent work of H. H. Chan, W.-C. Liaw, K. P. Loo and the author.
Date received: October 28, 2008
Local and global zeros of ternary quadratic forms
by
John Friedlander
University of Toronto
Coauthors: Henryk Iwaniec
We study a problem of Serre and variations thereof concerning the existence of non-trivial zeros of the Legendre quadratic form ax2 + by2 - z2.
Date received: October 28, 2008
Computing level one Hecke eigensystems (mod p)
by
Alexandru Ghitza
University of Melbourne
Coauthors: Craig Citro
We describe an algorithm for computing the complete list of systems of Hecke eigenvalues coming from modular forms (mod p) of level one. The focus will be on the theoretical underpinnings, but we will also touch upon the implementation of the algorithm in Sage.
Date received: October 28, 2008
Iwasawa theory of elliptic curves for supersingular primes
by
Byoung Du Kim
Victoria University of Wellington
I will introduce Iwasawa theory for elliptic curves. The focus will be on the new theory, ``the plus/minus Iwasawa theory'' for supersingular primes, and I will explain how it is used to solve some problems including the parity conjecture.
Date received: November 3, 2008
A fundamental system of units for a family of algebraic number fields of degree 12
by
Claude Levesque
U. Laval, Quebec
Let a and b be the two real roots of the
(assumed irreducile) polynomial X2+DX +d where
D, d are integers such that d divides D and D2 -4d > 0.
Let K be the sextic field Q( w) where
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Date received: October 28, 2008
On the spacings between C-nomial coefficients
by
Florian Luca
Mathematical Institute, UNAM, Morelia
Coauthors: Pantelimon Stanica, Diego Marques
Let (Cn)n ≥ 0 be the Lucas sequence Cn+2=aCn+1+bCn for all n ≥ 0, where C0=0 and C1=1. For 1 ≤ k ≤ m-1 let
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Date received: October 28, 2008
Rational-derived polynomials
by
Jim MacDougall
University of Newcastle
A polynomial in Q[x] is called rational-derived if all of its roots are rational and all the roots of all of its derivatives are rational. All such polynomials of degree 3 are easily characterised. Surprisingly, already at degree 4 we do not know how to describe all such polynomials. In particular, no rational-derived quartic with 4 distinct roots has been found. This talk will survey what is known about this and some related problems.
Date received: October 24, 2008
Inequities in the Shanks-Rényi prime number race
by
Greg Martin
University of British Columbia
Coauthors: Daniel Fiorilli (Université de Montréal)
Let p(x;q, a) denote the number of primes up to x that are congruent to a (mod q). It has been well-observed that an inequality of the type p(x;q, a) > p(x;q, b) is more likely to hold if a is a nonsquare modulo q and b is a square modulo q (the so-called “Chebyshev bias” in comparative prime number theory). However, the tendencies of the various p(x;q, a) (for nonsquares a) to dominate p(x;q, b) have different strengths: the asymptotic (logarithmic) density of the real numbers x for which p(x;q, a) > p(x;q, b) can depend on a and b as well as on q. Some of these densities have been computed, but only by using laborious numerical integration of functions involving zeros of the appropriate Dirichlet L-functions. We present an asymptotic formula for these densities, which in its most explicit form explains which nonsquares a are most dominant for a given square b.
Date received: October 21, 2008
Modular curves of genus 3
by
Roger Oyono
University of French Polynesia
Coauthors: Enrique Gonzalez Jimenez
We present a method for computing equations of non-hyperelliptic modular curves (defined over the rationals) of genus 3.
Date received: October 30, 2008
Perfect powers expressible as sums of two cubes
by
Samir Siksek
University of Warwick
Coauthors: Imin Chen
We attack the equation a3+b3=cn (a, b, c coprime integers) using a combination of the modular approach (via Galois representations and modular forms) together with an obstruction to solutions which is of Brauer-Manin type. We solve this equation for a set of prime exponents n having Dirichlet density 0.628.
Date received: October 21, 2008
Cubic Thue equations with many solutions
by
Cameron Stewart
University of Waterloo
Let F be a cubic binary form with non-zero discriminant and integer coefficients. We shall show how that there is a positive number c, which depends on F, such that the Thue equation F(x, y)=m has at least c(logm)(1/2) solutions in integers x and y for infinitely many positive integers m.
Date received: November 2, 2008
Application of a theorem of Akhtari to families of quartic diophantine equations.
by
Gary Walsh
University of Ottawa
We elaborate on a recent theorem due to Shabnam Akhtari, and describe the applicability of this theorem to solving certain families of classical diophantine equations, and extensions thereof.
Date received: September 11, 2008
A fundamental cryptographic(?) algorithm
by
Hugh Williams
University of Calgary, Canada
Cryptosystems which rely for their security on the presumed difficulty of solving the discrete logarithm problem in quadratic number fields execute somewhat more slowly than the standard Diffie-Hellman or RSA techniques. Although this gap has narrowed somewhat in the last several years, in order to narrow it further, there is still a fundamental difficulty that must be addressed. This is the fast implementation of the operation of finding a reduced ideal equivalent to the product of two given ideals, the operation analogous to that of modular multiplication in rational number theory. In 1988 Daniel Shanks described an algorithm, which he called NUCOMP, for performing this operation. The beauty of this algorithm is that it does not require the large intermediate numbers that are needed after the usual ideal multiplication, which is subsequently followed by a reduction procedure. Although Shanks’ version of NUCOMP was developed for imaginary quadratic fields, van der Poorten was able to show that it could also be used for real quadratic fields. In this talk, I will describe the most recent version of NUCOMP, and present an analysis of why and how well it works.
Date received: October 30, 2008
On finiteness of odd superperfect numbers
by
Tomohiro Yamada
Department of Mathematics, Kyoto University
Some new results concerning the equation s(N)=aM, s(M)=bN are proved, which implies that there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.
Date received: October 28, 2008
On Hecke eigenvalues at Piatetski-Shapiro primes
by
Liangyi Zhao
Nanyang Technological University
Coauthors: Stephan Baier
Let l(n) be the normalized n-th Fourier coefficient of holomorphic cusp form for the full modular group. We show that for some constant C > 0 depending on the cusp form and every fixed c in 1 < c < 8/7, the mean value of l(p) is O ( exp( -C √{logN} )) as p runs over all (Piatetski-Shapiro) primes of the form [nc] with some natural number n ≤ N.
Date received: September 8, 2008