Finiteness of fourth-power exceptions

Prove that the set of positive integers that are neither fourth powers nor sums of an odd prime and a positive fourth power is finite, and determine or bound its size.

Background

The authors report 215,414,496 such exceptions below 1010 and additional exceptions as large as 1014, without an exhaustive search. They therefore leave the global finiteness and the magnitude of the exceptional set unresolved.

References

We would conjecture that this set is finite, but we are not sure how large it might be.

— Computational results on sums of a prime with squares or cubes  (2609.20505 - Applegate et al., 17 Sep 2026) in Section 4.1, final paragraph