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- W2783205753 abstract "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 $k geq 2$, there exists a positive integer $W(k)$ such that every sufficiently large multiple of $E_k := gcd(2^k - 1, k)$ 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 $gcd$ of the binary $k$'th powers is $E_k$.) Also, we explain how our results can be extended to arbitrary integer bases $b > 2$." @default.
- W2783205753 created "2018-01-26" @default.
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- W2783205753 date "2018-01-13" @default.
- W2783205753 modified "2023-09-27" @default.
- W2783205753 title "Waring's Theorem for Binary Powers" @default.
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