No 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.Communicated by Kannan Soundararajan. Michael A. Bennett is supported by NSERC. Adela Gherga and Samir Siksek are supported by an EPSRC Grant EP/S031537/1 "Moduli of elliptic curves and classical Diophantine problems".

We show that if n ≥ 3 n \geq 3 is a fixed integer, then there exists an effectively computable constant c ( n ) c (n) such that if x x , y y , and m m are integers satisfying x m − 1 x − 1 = y n − 1 y − 1 , y > x > 1 , m > n , \begin{equation*} \frac {x^m-1}{x-1} = \frac {y^n-1}{y-1}, \; \; y>x>1, \; m > n, \end{equation*} with gcd ( m − 1 , n − 1 ) > 1 \gcd (m-1,n-1)>1 , then max { x , y , m } > c ( n ) \max \{ x, y, m \} > c (n) . In case n ∈ { 3 , 4 , 5 } n \in \{ 3, 4, 5 \} , we solve the equation completely, subject to this non-coprimality condition. In case n = 5 n=5 , our resulting computations require a variety of innovations for solving Ramanujan-Nagell equations of the shape f ( x ) = y n f(x)=y^n , where f ( x ) f(x) is a given polynomial with integer coefficients (and degree at least two), and y y is a fixed integer.