Updates on Lehmer's Totient Problem

Question

As I read here and in many books on the Theory of Numbers, we are yet to prove or disprove the existence of any composite $n$ such that $\phi(n)\mid n-1$. Is there progress in this line?

Answer 1

Depends what you call progress.

Grau Ribas and Luca, Cullen numbers with the Lehmer property, Proc. Amer. Math. Soc. 140 (2012), no. 1, 129–134, MR2833524 (2012e:11002), prove there are no counterexamples of the form $k2^k+1$.

Burcsi, Czirbusz, and Farkas, Computational investigation of Lehmer's totient problem, Ann. Univ. Sci. Budapest. Sect. Comput. 35 (2011), 43–49, MR2894552, prove that if $n$ is composite and $k\phi(n)=n-1$ and $n$ is a multiple of 3 then $n$ has at least 40000000 prime divisors, and $n\ge10^{360000000}$.

There's more. If you have access to MathSciNet, just type in Lehmer and totient, and see what comes up.