The Scholz conjecture for n=2^m(23)+7, m ∈ ℕ^*

2023/02/04 by Amadou Tall, Tall, Amadou
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2302.02143

openalex publication_date 2023/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Scholz conjecture on addition chains states that ℓ(2ⁿ-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 \cite9 to prove that there are an infinity of integers satisfying ℓ(2n) = ℓ(n). Later on, Thurber \cite4 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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