Odd values of the Ramanujan tau function

Abstract

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.