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Hardy and Littlewood conjectured that every large integer $n$ that is not asquare 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 asymptoticallygiven by \mathcal{R}(n) \sim \frac{\sqrt{n}}{\logn}\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 aprime 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 Siegelzero, 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 largepositive integer and $0< \delta_1 \le 0.000025$.

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