1. CJM 2005 (vol 57 pp. 298)
 Kumchev, Angel V.

On the WaringGoldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers
We investigate exceptional sets in the WaringGoldbach problem. For
example, in the cubic case, we show that all but
$O(N^{79/84+\epsilon})$ integers subject to the necessary local
conditions can be represented as the sum of five cubes of primes.
Furthermore, we develop a new device that leads easily to similar
estimates for exceptional sets for sums of fourth and higher powers of
primes.
Categories:11P32, 11L15, 11L20, 11N36, 11P55 

2. 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 

3. 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 

4. CJM 1998 (vol 50 pp. 465)
 Balog, Antal

Six primes and an almost prime in four linear equations
There are infinitely many triplets of primes $p,q,r$ such that the
arithmetic means of any two of them, ${p+q\over2}$, ${p+r\over2}$,
${q+r\over2}$ are also primes. We give an asymptotic formula for
the number of such triplets up to a limit. The more involved
problem of asking that in addition to the above the arithmetic mean
of all three of them, ${p+q+r\over3}$ is also prime seems to be out
of reach. We show by combining the HardyLittlewood method with the
sieve method that there are quite a few triplets for which six of
the seven entries are primes and the last is almost prime.}
Categories:11P32, 11N36 
