Finiteness of partition numbers near kth powers

Establish that for every integer k > 1 and every real number d ≥ 0, only finitely many positive integers n satisfy A_k(n) ≤ d, where A_k(n) is the distance between p(n) and the nearest kth power.

Background

For each k > 1, the paper defines A_k(n) as the minimum distance between the partition number p(n) and an integer kth power. Numerical computations suggest that partition numbers are unusually far from perfect powers, apart from small exceptions.

Conjecture 1 formalizes this repulsion by asserting that any fixed distance threshold d can be attained by A_k(n) for only finitely many n.

References

Conjecture 1. If k > 1 and d ≥ 0, then there are at most finitely many n for which A_k(n) ≤ d.

— Do perfect powers repel partition numbers?  (2501.03754 - Merca et al., 7 Jan 2025) in Section “Further Conjectures: Do kth Powers Repel Partition Numbers?”, Conjecture 1

Based on numerics performed on a computer, we make the following conjecture. Conjecture 1. If k > 1 and d ≥ 0, then there are at most finitely many n for which A_k(n) ≤ d.

— Do perfect powers repel partition numbers?  (2501.03754 - Merca et al., 7 Jan 2025) in Section 2, “Further Conjectures: Do kth Powers Repel Partition Numbers?”, Conjecture 1