Mircea Merca1, Ken Ono2, Wei-Lun Tsai3
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
1 Department of Mathematical Models and Methods, Fundamental Sciences Applied in Engineering Research Center, National University of Science and Technology Politehnica Bucharest, 060042 Bucharest, Romania and Academy of Romanian Scientists, 050044 Bucharest, Romania, mircea.merca@upb.ro
2 Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA, ken.ono691@virginia.edu
3 Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA weilun@mailbox.sc.edu
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application, Volume 17 no 1, 2025
