A conjecture in addition chains related to Scholz’s conjecture

Abstract:

Let $l(n)$ denote, as usual, the length of a shortest addition chain for a given positive integer n. The most famous unsolved problem in addition chains is Scholz’s 1937 conjecture that for all natural numbers $n,l({2^n} - 1) \leq l(n) + n - 1$. While this conjecture has been proved for certain classes of values of n, its validity for all n is yet an open problem. In this paper, we put forth a new conjecture, namely, that for each integer $n \geq 1$ there exists an addition chain for ${2^n} - 1$ whose length equals $l(n) + n - 1$. Obviously, our conjecture implies (and is stronger than) Scholtz’s conjecture. However, it is not as bold as conjecturing that $l({2^n} - 1) = l(n) + n - 1$, which is known to hold, so far, for only the twenty-one values of n which were obtained by Knuth and Thurber after extensive computations. By utilizing a series of algorithms we establish our conjecture for all $n \leq 128$ by actually computing the desired addition chains. We also show that our conjecture holds for infinitely many n, for example, for all n which are powers of 2.