  Hardy and Littlewood conjectured that every large integer $n$ that is not a
square is the sum of a prime and a square. They believed that the number
$\mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically
given by \mathcal{R}(n) \sim \frac{\sqrt{n}}{\log
n}\prod_{p=3}^{\infty}(1-\frac{1}{p-1}(\frac{n}{p})), where $p$ is a prime, $m$
is an integer, and $(\frac{n}{p})$ denotes the Legendre symbol. Unfortunately,
as we will later point out, this conjecture is difficult to prove and not
\emph{all} integers that are nonsquares can be represented as the sum of a
prime and a square. Instead in this paper we prove two upper bounds for
$\mathcal{R}(n)$ for $n \le N$. The first upper bound applies to \emph{all} $n
\le N$. The second upper bound depends on the possible existence of the Siegel
zero, and assumes its existence, and applies to all $N/2 < n \le N$ but at most
$\ll N^{1-\delta_1}$ of these integers, where $N$ is a sufficiently large
positive integer and $0< \delta_1 \le 0.000025$.

Number Theory