  We prove a number of results regarding odd values of the Ramanujan
$\tau$-function. For example, we prove the existence of an effectively
computable positive constant $\kappa$ such that if $\tau(n)$ is odd and $n \ge
25$ then either \[
  P(\tau(n)) \; > \; \kappa \cdot \frac{\log\log\log{n}}{\log\log\log\log{n}}
\] or there exists a prime $p \mid n$ with $\tau(p)=0$. Here $P(m)$ denotes the
largest prime factor of $m$. We also solve the equation $\tau(n)=\pm 3^{b_1}
5^{b_2} 7^{b_3} 11^{b_4}$ and the equations $\tau(n)=\pm q^b$ 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 Gy\H{o}ry, and the modular approach via Galois representations of
Frey-Hellegouarch elliptic curves.

Number Theory