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?


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.

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    $\begingroup$ Sorry, I don't have access to MathSciNet. $\endgroup$ Jul 23 '12 at 5:17
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    $\begingroup$ @labbhattacharjee You can also try scholar.google.com and other serach engines. $\endgroup$
    – Tommi
    Oct 30 '14 at 11:29

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