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

Abstract

The Carmichael lambda function $\lambda (n)$ is defined to be the smallest positive integer $m$ such that $a^m\equiv 1(\mathrm{mod}n)$ for all $(a,n)=1.$ ${\lambda }_k(n)$ is defined to be the $k$th iterate of $\lambda (n).$ Let L(n) be the smallest $k$ for which ${\lambda }_k(n)=1.$ It's easy to show that $L(n)\ll \log n.$ It's conjectured that $L(n)\asymp \log \log n,$ but previously it was not known to be $o(\log n)$ for almost all $n.$ We will show that $L(n)\ll (\log n{)}^{\delta }$ for almost all $n,$ for some $\delta <1.$ We will also show $L(n)\gg \log \log n$ for almost all $n$ and conjecture a normal order for L(n).

9 pages

References in corpus (1)

• The iterated Carmichael λ-function and the number of cycles of the power generator