The number of iterates of the Carmichael lambda function required to reach 1

2012/03/21 by Nick Harland, Harland, Nick
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1203.4791

9 pages

arxiv created 2012/03/21 · openalex publication_date 2012/03/21 · arxiv updated 2012/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Carmichael lambda function λ(n) is defined to be the smallest positive integer m such that a^m ≡ 1 \pmodn for all (a,n)=1. λ_k(n) is defined to be the kth iterate of λ(n). Let L(n) be the smallest k for which λ_k(n)=1. It's easy to show that L(n) ≪ log n. It's conjectured that L(n)\asymp loglog n, but previously it was not known to be o(log n) for almost all n. We will show that L(n) ≪ (log n)^δ for almost all n, for some δ<1. We will also show L(n) ≫ loglog n for almost all n and conjecture a normal order for L(n).

Related

• The iterated Carmichael lambda function
• Runs of integers with constant values of the Carmichael function
• The Asymptotic Behavior of Compositions of the Euler and Carmichael Functions
• Carmichael numbers and least common multiples of p-1
• On Generalized Carmichael Numbers