Mircea Merca, Ken Ono, Wei-Lun Tsai
Abstract. In 2013 Zhi-Wei Sun conjectured that p(n) is never a power of an integer when n > 1. We confirm this claim in many cases. We also observe that integral powers appear to repel the partition numbers. If k > 1 and ∆k(n) is the distance between p(n) and the nearest kth power, then for every d ≥ 0 we conjecture that there are at most finitely many n for which ∆k(n) ≤ d. More precisely, for every ε > 0, we conjecture that
Mk(d) := max{n : ∆k(n) ≤ d} = o(d∆ε).
In k-power aspect with d fixed, we also conjecture that if k is sufficiently large, then
Mk(d) = max {n : p(n) − 1 ≤ d} .
In other words, 1 generally appears to be the closest kth power among the partition numbers.
Keywords: partition function, perfect powers.
MSC: 11P82, 05A17, 05A20.
DOI 10.56082/annalsarscimath.2025.1.95
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics on Its Application, Volume 17 no 1, 2025
