The Scholz conjecture for n=2^m(23)+7, m ∈ℕ^*
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The Scholz conjecture on addition chains states that ℓ(2^n-1) ≤ℓ(n) + n -1 for all integers n where ℓ(n) stands for the minimal length of all addition chains for n. It is proven to hold for infinite sets of integers. In this paper, we will prove that the conjecture still holds for n=2^m(23)+7. It is the first set of integers given by Thurber <cit.> to prove that there are an infinity of integers satisfying ℓ(2n) = ℓ(n). Later on, Thurber <cit.> give a second set of integers with the same properties (n=2^2m+k+7 + 2^2m+k+5 + 2^m+k+4 + 2^m+k+3 + 2^m+2 + 2^m+1 + 1). We will prove that the conjecture holds for them as well.
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