[Resource Topic] 2023/020: The Scholz conjecture on addition chain is true for infinitely many integers with ℓ(2n) = ℓ(n)

Welcome to the resource topic for 2023/020

Title:
The Scholz conjecture on addition chain is true for infinitely many integers with ℓ(2n) = ℓ(n)

Authors: Amadou TALL

Abstract:

It is known that the Scholz conjecture on addition chains is true for all integers n with ℓ(2n) = ℓ(n) + 1. There exists infinitely many integers with ℓ(2n) ≤ ℓ(n) and we don’t know if the conjecture still holds for them. The conjecture is also proven to hold for integers n with v(n) ≤ 5 and for infinitely many integers with v(n) = 6. There is no specific results on
integers with v(n) = 7. In [14], an infinite list of integers satisfying ℓ(n) = ℓ(2n) and v(n) = 7
is given by Thurber. In this paper, we prove that the conjecture holds for all of them.

ePrint: https://eprint.iacr.org/2023/020

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