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SUMMARY:Florian Luca (University of the Witwatersrand\, South Africa)
DTSTART:20200604T130000Z
DTEND:20200604T132500Z
DTSTAMP:20260423T011218Z
UID:CANT2020/43
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2020/43/
 ">Prime factors of the Ramanujan $\\tau$-function</a>\nby Florian Luca (Un
 iversity of the Witwatersrand\, South Africa) as part of Combinatorial and
  additive number theory (CANT 2021)\n\n\nAbstract\nLet $\\tau(n)$ be the R
 amanujan $\\tau$-function of $n$. \nIn this talk\, we prove some results a
 bout prime factors of $\\tau(n)$ and its iterates. \nAssuming the Lehmer c
 onjecture that $\\tau(n)\\ne 0$ for all $n$\, \nwe show that if $n$ is eve
 n and $k\\ge 1$\, then $\\tau^{(k)}(n)$ is divisible \nby a prime $p\\ge 3
 ^{k-1}+1$. \nGiven a fixed finite set of odd primes $S=\\{p_1\,\\ldots\,p_
 \\ell\\}$\, \nwe give a bound on the number of solutions of $n$ of the equ
 ation \n$\\tau(n)=\\pm p_1^{a_1}\\cdots p_\\ell^{a_\\ell}$ for integers $a
 _1\,\\ldots\,a_\\ell$\nand in case $S:=\\{3\,5\,7\\}$\, we show that there
  is no such $n>1$.  \n\nJoint work with S. Mabaso and P. Stӑnicӑ.\n
LOCATION:https://researchseminars.org/talk/CANT2020/43/
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