  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 $b > 2$.
