CECM Colloquium

**Wednesday February 21, 2001 From
3:30 - 4:30 in K9509, SFU
**

**
**

** Stephen Choi, Simon Fraser University
Talks on
**

**Small Prime Solutions of Quadratic Equations
**

** Abstract:**
In this talk, we will study the small prime solutions of certain diophantine
equations by using the Hardy-Littlewood (circle) method. In particular, we
prove the following. If $b_1,\cdots ,b_5$ are non-zero integers, then the
quadratic equation $b_1p_1^2+\cdots +b_5p_5^2=n$ has prime solutions
satisfying $p_j \ll \sqrt{|n|}+\max\{|b_j|\}^{20+\epsilon}$. In contrast
to the earlier works which treat the enlarged major arc by the
Deuring-Heilbronn phenomenon about the Siegel zero. We will explain the
possible existence of Siegel zero does not have special influence and hence
the Deuring-Heilbronn phenomenon can be avoided. This observation enables us to
get better results without numerical computations.