Odd values of the Ramanujan tau function

We prove a number of results regarding odd values of the Ramanujan τ-function. For example, we prove the existence of an effectively computable positive constant κ such that if τ(n) is odd and n≥ 25 then either P(τ(n))>κ·logloglognloglogloglognor there exists a prime p∣ n with τ(p) = 0. Here P(m) denotes the largest prime factor of m. We also solve the equation τ(n)=±3b15b27b311b4 and the equations τ(n) = ± q^b where 3 ≤ 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.