On powers that are sums of consecutive like powers

1 Background The problem of cubes that are sums of consecutive cubes goes back to Euler ([10] art. 249) who noted the remarkable relation 33 + 43 + 53 = 63. Similar problems were considered by several mathematicians during the nineteenth and early twentieth century as surveyed in Dickson’sHistory of the Theory of Numbers ([7] p. 582–588). These questions are still of interest today. For example, both Cassels [5] and Uchiyama [17] determined the squares that can be written as sums of three consecutive cubes. Stroeker [16] determined all squares that are expressible as the sum of 2 ≤ d ≤ 50 consecutive cubes, using a method based on linear forms in elliptic logarithms. More recently, Bennett, Patel and Siksek [2] determined all perfect powers that are expressible as sums of 2 ≤ d ≤ 50 consecutive cubes, using linear forms in logarithms, sieving and Frey curves. There has been some interest in powers that are sums of k-th powers for other exponents k . For example, the solutions to the equation