Let $p> 2$ be an odd prime. For each integer $b$ with $1\leq b<p$ and $(b, p)=1$, there is a unique integer $c$ with $1\leq c<p$ such that $bc\equiv 1(\bmod p)$. Let $M(1,p)$ denote the number of solutions $(b,c)$ of the congruence equation $bc\equiv 1(\bmod p)$ with $1\leq b,c<p$ such that $b,c$ are of opposite parity. D. H. Lehmer (see problem F12 of [Unsolved Problems in Number Theory, 1994, page 251]) posed the problem to find $M(1,p)$ or at least to say something non-trivial about it.
In this talk, the research history since 1993 will be shown. And various generalizations are also given.