Runs of integers with constant values of the Carmichael function

Abstract

In 2023, the first author and Vandehey proved that the largest $k$ for which the string of equalities $\lambda (n+1)=\lambda (n+2)=\cdots =\lambda (n+k)$ holds for some $n\le x$, where $\lambda$ is the Carmichael $\lambda$ function, is bounded above by $O\left((\log x\log \log x{)}^2\right)$. Their method involved bounding the value of $\lambda (n+i)$ from below using the prime factorization of $n+i$ for each $i\le k$. They then used the fact that every $\lambda (n+i)$ had to satisfy this bound. Here we improve their result by incorporating a reverse counting argument on a result of Baker and Harman on the largest prime factor of a shifted prime.