REPRESENTATION OF EVEN INTEGERS AS THE SUM OF A PRIME NUMBER AND A SQUARE

Abstract

We study the representation of even integers in the form $n=p+k^2$, where $p$ is a prime number and $k$ is an odd positive integer. After establishing the necessary modular constraints, we formulate a conjecture asserting that every sufficiently large such $n$ admits at least one such representation. We provide heuristic arguments based on the prime number theorem, numerical evidence up to $2 \times 10^6$, and discuss possible asymptotic formulas using the Hardy-Littlewood circle method. Several counterexamples are identified, and their structures are analyzed.