A note on the Aiello–Subbarao conjecture on addition chains

Abstract

Given a positive integer $x$, an addition chain for $x$ is an increasing sequence of positive integers $1=c_0,c_1, \ldots , c_n=x$ such that for each $1\leq k\leq n,$ $c_k=c_i+c_j$ for some $0\leq i\leq j\leq k-1$. In 1937, Scholz conjectured that <em>for each positive integer</em> $x$, $\ell(2^x-1) \leq \ell(x)+ x-1,$ where $\ell(x)$ denotes the minimal length of an addition chain for $x.$ In 1993, Aiello and Subbarao stated the apparently stronger conjecture that <em>there is an addition chain for</em> $2^x-1$ <em>with length equals to</em> $\ell(x)+x-1 .$ We note that the Aiello–Subbarao conjecture is not stronger than the Scholz (also called the Scholz–Brauer) conjecture.