On powers that are sums of consecutive like powers

2016/07/28 by Vandita Patel, Samir Siksek, Patel, Vandita +1
Mathematics · #11B61 #11D68 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B61 #msc:11D68

paper · pdf · doi:10.48550/arxiv.1607.08418

arxiv created 2016/07/28 · arxiv updated 2016/07/29

Abstract

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 equation x^k+(x+r)^k+⋯+(x+(d-1)r)^k=yⁿ, x,~y,~n ∈ ℤ, n ≥ 2, has no solutions.

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