Repellent properties of perfect powers on partition functionsMPIM

by Swati (MPIM)

Wednesday Nov 18, 2026, 3:30 PM → 4:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Number theory lunch seminar

n 2013, Sun conjectured that the partition function $p(n)$ is never a perfect power for $n \geq 2$. Building on this, Merca, Ono, and Tsai recently observed that for any fixed integers $d \geq 0$ and $k \geq 2$, there appear to be only finitely many integers $n$ such that $p(n)$ differs from a perfect $k$th power by at most $d$. Denoting by $M_k(d)$ the largest such $n$, they conjectured that $M_k(d) = o(d^{\epsilon})$ for every $\epsilon > 0$.
In this talk, we establish sharp lower bounds and provide heuristics which suggest that $M_k(d)$ in fact grows polylogarithmically in $d$, i.e. of order $(\log d)/(\log 2)$ . Further, we show 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 $1$. This is a joint work with Summer Haag, Praneel Samanta, Holly Swisher, Stephanie Treener, and Robin Visser.