Prime factors of the Ramanujan $\tau$-function

Abstract: Let $\tau(n)$ be the Ramanujan $\tau$-function of $n$. In this talk, we prove some results about prime factors of $\tau(n)$ and its iterates. Assuming the Lehmer conjecture that $\tau(n)\ne 0$ for all $n$, we show that if $n$ is even and $k\ge 1$, then $\tau^{(k)}(n)$ is divisible by a prime $p\ge 3^{k-1}+1$. Given a fixed finite set of odd primes $S=\{p_1,\ldots,p_\ell\}$, we give a bound on the number of solutions of $n$ of the equation $\tau(n)=\pm p_1^{a_1}\cdots p_\ell^{a_\ell}$ for integers $a_1,\ldots,a_\ell$ and in case $S:=\{3,5,7\}$, we show that there is no such $n>1$.

Joint work with S. Mabaso and P. Stӑnicӑ.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.

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