School of Mathematics and Statistics |
Department of Mathematics and Statistics |

**NUMBER THEORY SEMINAR**

**Fall 2005/ Winter 2006**

In particular, I would like to show the importance of Selberg's sieve in prime number theory. My goal is

to expose the recent groundbreaking work of Goldston, Pintz, and Yildirim concerning small gaps between

primes. In the second talk, I will present an outline of their proof that there are infinitely many consecutive

primes with very small gaps between them. In fact,their approach presents a promising method for attacking

the twin prime conjecture.

Let p_n be the n-the prime number. We show that liminf (p_(n+1)-p_n)/log(p_n) =0. I will discuss how this

proof uses an upper bound sieve and the Bombieri-Vinogradov theorem. Moreover, we discuss how the method

applies to finding bounded gaps between primes assuming certain distribution properties of primes in arithmetic progressions.

does not vanish, a modular form of weight 3/2 whose Fourier coefficients relate to the central values

of the L-series of the imaginary quadratic twists of f, giving an explicit formula for such central values.

We will present joint work with Z. Mao and F. Rodriguez-Villegas which generalize Gross's algorithm and

formula to include the case L(f,1)=0, and to the case of the real quadratic twists of f.

points on elliptic curves of the form $y^2=x^3 \pm nx$, with $n$ a prime power.

Dirichlet box principle shows that the uniform exponent of simultaneous approximation of xi, xi^2 and xi^3

by rational numbers with the same denominator is at least 1/3 ~ 0.3333. In this talk, we show that this

exponent is at most the smallest positive root lambda ~ 0.4245 of the polynomial X^2-(2*gamma+1)X+gamma

where gamma denotes the golden ratio.

torsion subgroup, roughly speaking, has as much Galois symmetry as is

possible. I will discuss a theorem that, when counted according to

height, almost all elliptic curves are Serre curves. If time permits,

I'll discuss an application of this theorem to a question of averages of

the constants in the Lang-Trotter conjecture.

groups, S. L. Ma proposed the following conjecture: Ma's conjecture.

Let p be an odd prime and b, m, r be positive integers. Then (1): x^2

= 2^(2b+2)p^(2m) ? 2^(b+1)p^(m+r) + 1 holds with some positive integer

x if and only if p = 5, b = 3, m = 1 and r = 2 (when x = 49). While we

cannot prove Ma's conjecture, in my talk I will show that if p is

fixed, then the diophantine equation (1) has at most 2^30,000 positive

integer solutions (x, b, m, r). The proof uses the Subspace Theorem

and results on S-unit equations. This is joint work with Pantelimon Stanica.

Nathan Ng

nng362 at science.uottawa.ca

Tel. 613-562-5800 ext 3515

or

Damien Roy

droy at uottawa.ca

Tel. 613-562-5800 ext 3504