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.
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.
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