2025/01/07 by Mircea Merca, Ken Ono, Merca, Mircea +3 · 2 citations
Mathematics · #05A17. 05A20 #11P82 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2501.03754
openalex publication_date 2025/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Δk(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 Δk(n)≤ d. More precisely, for every ε>0, we conjecture that Mk(d):=max\n : Δk(n)≤ d\=o( dε). 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.