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 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 R(n) for n ≤ N. The first upper bound applies to all n ≤ 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 ≤ N but at most N^{1-delta_1} of these integers, where N is a sufficiently large positive integer and 0 delta1 ≤ 0.000025.