Abstract: Let n ≥3 be an odd number and contain a primitive root, for each 1≤a < n with ( a, n ) = 1, It is clear that there exists one and only one 1  a < n such that a a ≡1(mod n) . Such number a is called as lehmer D H number if a and  a are of opposite parity. This paper is to study the dist ribution property of difference between lehmer D H number and its inverse by use the estimation of kloos terman’s sum. and given an asymptotic formula.

Key words: Lehmer D H number, primitive root, inverse, distribution property, asymptotic formula