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SUMMARY:Repellent properties of perfect powers on partition functions [MPI
 M]
DTSTART:20261118T143000Z
DTEND:20261118T150000Z
DTSTAMP:20261112T153900Z
UID:indico-event-1785@math-events.uni-bonn.de
DESCRIPTION:Speakers: Swati (MPIM)\n\nNumber theory lunch seminar\nn 2013\
 , Sun conjectured that the partition function $p(n)$ is never a perfect po
 wer for $n \\geq 2$. Building on this\, Merca\, Ono\, and Tsai recently ob
 served that for any fixed integers $d \\geq 0$ and $k \\geq 2$\, there app
 ear 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 $\\epsil
 on > 0$.In this talk\, we establish sharp lower bounds and provide heurist
 ics 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 perfec
 t power is finite with probability $1$. This is a joint work with Summer H
 aag\, Praneel Samanta\, Holly Swisher\, Stephanie Treener\, and Robin Viss
 er.\n\nhttps://math-events.uni-bonn.de/event/1785/
LOCATION:MPIM\, Vivatsgasse\,  7 - Lecture Hall (Max Planck Institute for 
 Mathematics)
URL:https://math-events.uni-bonn.de/event/1785/
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