Prime plus one positive cube

Prove that every nonscube integer n greater than 78526384 can be represented as n=p+x^3, where p is an odd prime and x is a positive integer.

Background

This conjecture is the cube counterpart of the prime-plus-one-square conjecture and is used as the computational basis for bootstrapping toward two cubes. The authors verify the representation up to 1012 outside a finite computed exceptional set, but do not establish the stated global bound.

References

If $n>78526384$ is an integer that is not a cube, then $n=p+x3$, 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 1.5, Section 1

We conjecture that $\mathcal{E}_3$ is the complete list of positive, non-cube integers that cannot be written as the sum of an odd prime and a positive cube.

— Computational results on sums of a prime with squares or cubes  (2609.20505 - Applegate et al., 17 Sep 2026) in Section 3, opening paragraph