  A composite number $n$ is called a Lehmer number when $\phi(n) | n - 1$,
where $\phi$ is the Euler totient function. Lehmer's totient problem asks if
there exist any composite numbers $n$ such that $\phi(n)| n-1$? No such numbers
are known. In this paper we establish an Euler Totient Inequality and relate
some new parameters to Lehmer numbers. As an application of what we have done,
we show that for all prime numbers $p$ and for all odd square-free numbers $n$,
at most one of $n$ or $pn$ is a Lehmer number. Finally we suggest some open
problems for the future investigations on the Lehmer numbers.
