On the D.H. Lehmer problem

Abstract: 

Let p>2p>2 be an odd prime. For each integer bb with 1≤b<p1≤b<p and (b,p)=1(b,p)=1, there is a unique integer cc with 1≤c<p1≤c<p such that bc≡1(modp)bc≡1(modp). Let M(1,p)M(1,p) denote the number of solutions (b,c)(b,c) of the congruence equation bc≡1(modp)bc≡1(modp) with 1≤b,c<p1≤b,c<p such that b,cb,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)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.