Prime plus one positive square

Prove that every nonsquare integer n greater than 21679 can be represented as n=p+x^2, where p is an odd prime and x is a positive integer.

Background

The conjecture is used as the computational input for the paper's bootstrapping argument toward sums of a prime and two squares. The paper verifies the relevant representation up to 1014 outside an explicitly listed exceptional set, but does not prove that the bound 21679 is globally sufficient.

References

If $n>21679$ is an integer that is not a square, then $n=p+x2$, where $p$ is an odd prime and $x$ is a positive integer.

— Computational results on sums of a prime with squares or cubes  (2609.20505 - Applegate et al., 17 Sep 2026) in Conjecture 1.2, Section 1

We conjecture that $\mathcal{E}_2$ is the complete list of positive non-square integers that cannot be written as the sum of an odd prime and a positive square.

— Computational results on sums of a prime with squares or cubes  (2609.20505 - Applegate et al., 17 Sep 2026) in Section 2, immediately after equation (2.1)