Waring's Theorem for Binary Powers

TL;DR

This paper establishes an analogue of Waring's theorem for binary k'th powers, showing that large multiples of a specific gcd can be expressed as sums of a bounded number of such powers, with extensions to other bases.

Contribution

It proves a Waring-type theorem for binary k'th powers, identifying the minimal number of terms needed and extending results to arbitrary bases greater than 2.

Findings

Every sufficiently large multiple of gcd(2^k - 1, k) can be expressed as a sum of bounded binary k'th powers.

The gcd of all binary k'th powers is exactly gcd(2^k - 1, k).

Results can be generalized to arbitrary integer bases b > 2.

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 $b > 2$.