Some results for some conjectures in addition chains

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 greater than or equal to 1, a(i) = a(j) + a(k) for some 0 less than or equal to j, k < i. The smallest length r for which an addition chain for n exists is denoted by l(n), Scholz conjectured that l(2(n) - 1) :5 n + l(n) - 1. Aiello and Subbarao proposed a stronger conjecture which is "for each integer n greater than or equal to 1, there exists an addition chain for 2(n) - 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 MB-chain, we conjecture that it is equivalent to l(o)-chain and we prove that this conjecture is true for integers n less than or equal to 8 X 10(4). Also, we prove that the Scholz and Aiello-Subbarao conjectures are true for integers n less than or equal to 8 X 104.