1. CJM 2002 (vol 54 pp. 417)
 Wooley, Trevor D.

Slim Exceptional Sets for Sums of Cubes
We investigate exceptional sets associated with various additive
problems involving sums of cubes. By developing a method wherein an
exponential sum over the set of exceptions is employed explicitly
within the HardyLittlewood method, we are better able to exploit
excess variables. By way of illustration, we show that the number of
odd integers not divisible by $9$, and not exceeding $X$, that fail to
have a representation as the sum of $7$ cubes of prime numbers, is
$O(X^{23/36+\eps})$. For sums of eight cubes of prime numbers, the
corresponding number of exceptional integers is $O(X^{11/36+\eps})$.
Keywords:Waring's problem, exceptional sets Categories:11P32, 11P05, 11P55 

2. CJM 2002 (vol 54 pp. 71)
 Choi, KwokKwong Stephen; Liu, Jianya

Small Prime Solutions of Quadratic Equations
Let $b_1,\dots,b_5$ be nonzero integers and $n$ any integer. Suppose
that $b_1 + \cdots + b_5 \equiv n \pmod{24}$ and $(b_i,b_j) = 1$ for
$1 \leq i < j \leq 5$. In this paper we prove that
\begin{enumerate}[(ii)]
\item[(i)] if $b_j$ are not all of the same sign, then the above
quadratic equation has prime solutions satisfying $p_j \ll \sqrt{n}
+ \max \{b_j\}^{20+\ve}$; and
\item[(ii)] if all $b_j$ are positive and $n \gg \max \{b_j\}^{41+
\ve}$, then the quadratic equation $b_1 p_1^2 + \cdots + b_5 p_5^2 =
n$ is soluble in primes $p_j$.
\end{enumerate}
Categories:11P32, 11P05, 11P55 
