Finiteness of prime-square exceptions

Prove that only finitely many nonsquare positive integers cannot be represented as the sum of a prime and a positive square.

Background

The paper attributes this broader finiteness assertion to a Hardy–Littlewood conjecture. It motivates the finite exceptional-set approach used in the computational study of prime-plus-square representations.

References

Hardy and Littlewood conjectured that there are only finitely many non-square integers that cannot be written as a sum of a prime and a square (see Conjecture H).

— Computational results on sums of a prime with squares or cubes  (2609.20505 - Applegate et al., 17 Sep 2026) in Section 1, paragraph 7