Waring's Theorem for Binary Powers

01/13/2018

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by   Daniel M. Kane, et al.

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A natural number is a binary k'th power if its binary representation consists of k consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary k'th powers. More precisely, we show that for each integer k ≥ 2, there exists a positive integer W(k) such that every sufficiently large multiple of E_k := (2^k - 1, k) is the sum of at most W(k) binary k'th powers. (The hypothesis of being a multiple of E_k cannot be omitted, since we show that the of the binary k'th powers is E_k.) Also, we explain how our results can be extended to arbitrary integer bases b > 2.