An addition chain for a positive integer n is a sequence of positive in tegers 1 = a_{0}, a_{1}, …, a_{r} = n, such that for each 0 < i ≤ r, a_{i} = a_{j} + a_{k} for some 0 ≤ j, k < i. Let ℓ(n) be the smallest possible length of an addition chain for a positive integer n. Then Scholz conjectured that ℓ(2^{n}−1) ≤ n+ℓ(n)−1, which still remains open. It is known that the Scholz conjecture is true when ν(n) ≤ 4, whe re ν(n) is the number of 1’s in the binary representation of n. We give some properties of nonstar steps in acs and prove that the Scholz conjecture is true for infinitely many new integers including the case w here ν(n) = 5.