Straightedge-and-Compass Constructibility of the Reciprocal-$k$-th-Power Law

Abstract

Given positive lengths $a$ and $b$ and a positive integer $k$, let $c$ be determined by $$\frac{1}{c^k}=\frac{1}{a^k}+\frac{1}{b^k}.$$ We prove that a single finite unmarked-straightedge-and-compass construction producing $c$ for every pair $a,b$ exists if and only if $k$ is a power of two. Necessity follows by specializing to $a=b=1$ and applying the algebraic-degree obstruction to $2^{1/k}$; sufficiency is established by a recursive construction using third, fourth, and mean proportionals. We give an explicit construction for $k=4$ and its iteration for $k=8,16,\dots$. We also describe a plane construction, intersecting a line with a Lamé curve, that realizes the same relation geometrically for every real $k>1$, although generally not by classical straightedge-and-compass operations.

8 pages, 3 figures; includes an explicit construction for k=4 and a complete classification for positive integer k