Some Results for Some Conjectures in Addition Chains

Abstract

An addition chain for a positive integer n is a sequence of positive integers 1 = a ₀ < a ₁ <… < a _r = n, such that for each i ≥ 1, a _i = a _j + a _k for some 0 ≤ j, k < i. The smallest length r for which an addition chain for n exists is denoted by ℓ(n). Scholz conjectured that ℓ(2ⁿ − 1) ≤ n + ℓ(n) − 1. Aiello and Subbarao proposed a stronger conjecture which is “for each integer n ≥ 1, there exists an addition chain for 2ⁿ − 1 with length equals n+ℓ(n) − 1.” This paper improves Brauer’s result for the Scholz conjecture. We propose a special class of addition chain called M B-chain, we conjecture that it is equivalent to ℓ ^° -chain and we prove that this conjecture is true for integers n ≤ 8 × 10⁴. Also, we prove that the Scholz and Aiello-Subbarao conjectures are true for integers n ≤ 8 × 10⁴.