Number Theory

    

A Bound for the Number of Addition Chains of Arbitrary Length

Authors: Theophilus Agama

Define the function F(m, β(m) − m) as the number of integers n in the interval [2^m, 2^{m+1}) such that l(n) ≤ β(m). By applying ideas from De Koninck et al. (2025), we obtain the following general upper bound: F(m, β(m) − m) ≤ exp[ (β(m) − m) · (2·log β(m) + (1 + ε)·2·log log m + O(1)) ] for any small ε > 0 as m tends to infinity. This result generalizes a recent bound proved in the work of De Koninck and collaborators.

Comments: 6 Pages.

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Submission history

[v1] 2025-06-20 20:10:04

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