Abstract. Euler noted the relation 6 3 = 3 3 + 4 3 + 5 3 and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker's work by determining all perfect powers that are sums of at most 50 consecutive cubes. Our methods include descent, linear forms in two logarithms, and Frey-Hellegouarch curves.

Let k ≥ 2 be even, and let r be a non-zero integer. We show that for almost all d ≥ 2 (in the sense of natural density), the equationhas no solutions.

Using only elementary arguments, Cassels solved the Diophantine equation (x − 1) 3 + x 3 + (x + 1) 3 = z 2 (with x, z ∈ Z). The generalization (x − 1) k + x k + (x + 1) k = z n (with x, z, n ∈ Z and n ≥ 2) was considered by Zhongfeng Zhang who solved it for k ∈ {2, 3, 4} using Frey-Hellegouarch curves and their corresponding Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solution for k = 5 is x = z = 0, and that there are no solutions for k = 6. The chief innovation we employ is a computational one, which enables us to avoid the full computation of data about cuspidal newforms of high level.

We determine primitive solutions to the equation (x − r) 2 + x 2 + (x + r) 2 = y n for 1 ≤ r ≤ 5, 000, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.

We study the equation Fn + Fm = y p , where Fn and Fm are respectively the n-th and m-th Fibonacci numbers and p ≥ 2. We find all solutions under the assumption n ≡ m (mod 2).

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 describe a computationally efficient approach to resolving equations of the form C 1 x 2 + C 2 = y n in coprime integers, for fixed values of C 1 , C 2 subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier.

No abstract