The Problem Of The Mean Value Of The Difference Of The Integer Power Mod Q

The distribution of an integer and its inverse mod q is a current topic in number theory,many experts and scholars have studied it,a scries of impotant results have been obtianed.The paper introduce the distribution of the differ-ence of different integer power mod q,it is a generalization of the distribution of an integer and its inverse mod q.By using some methods in elementary number theory,analytic number theory,uniting the trigonometric sums and the estima-tion for the two-term exponential sums,the paper study the mean value of the difference of the integer power mod q and give interesting asymptotic formula.The results are as followsLet p be a prime and a be a positive integer,q =p?,m1,m2 be constants of different positive integers,0<?,?1,?2<1 be real constants,k be any non-negative integer.1.For any integer a with 1<a<q,(a,q)= 1,there exist unique integer b with 1 ? b? q such that b = am1(mod q),denote it by(am1)q.If q>[1/?],the paper study the distribution of the difference of the integer power mod q and get the asymptotic formula,as follows where ?' denotes a summation over all a such that(a,q)= 1,?>0 be any real constants,O denotes some constants with ?,m1,m2,k;2.If q>max{[1/?1],[1/?2]},the paper study a generalized D.H.Lehmer prob-lem over incomplete intervals,give an interesting asymptotic formula where O denotes some constants with ?1,?2,m1,m2,k.