Odd values of the Ramanujan tau function

Abstract

We prove a number of results regarding odd values of the Ramanujan $\ddot{I}„$-function. For example, we prove the existence of an effectively computable positive constant $\hat{I}º$ such that if $\ddot{I}„(n)$ is odd and $n\ge 25$ then either \[ P(τ(n)) \; > \; κ\cdot \frac{\log\log\log{n}}{\log\log\log\log{n}} \] or there exists a prime $p\mid n$ with $\ddot{I}„(p)=0$. Here $P(m)$ denotes the largest prime factor of $m$. We also solve the equation $\ddot{I}„(n)=\pm 3^{b_1}{5}^{b_2}{7}^{b_3}11^{b_4}$ and the equations $\ddot{I}„(n)=\pm q^b$ where $3\le q<100$ is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue--Mahler equations due to Bugeaud and Győry, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves.