Odd values of the Ramanujan tau function

Abstract

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 eitherlog log log n log log log log n or 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) = ±3 b 1 5 b 2 7 b 3 11 b 4 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.