upper bounds on the solutions to n = p+m^2

ardy 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 begin{equation*} mathcal{r}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), end{equation*} where $p$ is a prime, $m$ is an integer, and $left(frac{n}{p}right)$ 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