| Ken Ono†
Abstract: A conjecture by Sun states that the partition function p(n), for n > 1, is never a perfect power. Recent work by Merca et al. proposes generalizations of perfect-power repulsion for p(n). In this note, we prove these generalizations for the functions pB(n) which count the number of partitions of n with the largest part ≤ B. If B ≥ 4 and k ≥ 3 with k ∤ (B – 1), then we prove that there are only finitely many pairs (n, m) for which |pB(n) – mk| ≤ d. These results support Sun and Merca et al.’s conjectures, as pB(n) → p(n) when B → +∞. To prove this, we reduce the problem to Siegel’s Theorem, which guarantees the finiteness of integral points on curves with genus ≥ 1. Keywords: partition function, perfect powers. MSC: 11P82, 05A17, 05A20. DOI 10.56082/annalsarscimath.2026.2.31 †ken.ono691@virginia.edu, Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA and Academy of Romanian Scientists, 050044 Bucharest, Romania |
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application,
ISSN ONLINE 2066 – 6594
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