2026/01/26 by Summer Haag, Praneel Samanta, Swati +3 · 1 voice
Mathematics · #math.NT
paper · pdf · doi:10.1007/s40687-026-00611-4
In 2013, Sun conjectured that the partition function p(n) is never a perfect power for n ≥ 2. Building on this, Merca, Ono, and Tsai recently observed that for any fixed integers d ≥ 0 and k ≥ 2, there appear to be only finitely many integers n such that p(n) differs from a perfect kth power by at most d. Denoting by Mk(d) the largest such n, they conjectured that Mk(d) = o(dε) for every ε> 0. In this paper, we investigate the asymptotic growth of analogs of Mk(d) for a wide class of partition functions. We establish sharp lower bounds and provide heuristics which suggest that Mk(d) in fact grows polylogarithmically in d, i.e. of order log2(d). More generally, we prove that if f(n) is a suitably random chosen function with asymptotic growth rate similar to that of p(n), then the set of integers n for which f(n) is a perfect power is finite with probability 1.