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1. CJM 2002 (vol 54 pp. 417)
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 Hardy-Littlewood 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)
Small Prime Solutions of Quadratic Equations Let $b_1,\dots,b_5$ be non-zero 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 |