The Distribution Of Residue Of Powers Of An Integer And Its Related Problems

The distribution of an element and its inverse modulo q is one of important problems of number theory,which brings about scholars’ wide further investi-gation profoundly.Therefore,there are many fantastic conclusions for above-mentioned problem.Our goals are to study the maximal difference of powers of an element modulo q,the residue of powers of an integer modulo q and an inter-esting generalization of Lehmer problem.The main tools in this paper are the elemental methods、the properties of exponential sums and exponential sums estimate with sparse rational function.And we obtain the concrete contents in the following:1.Let q≥ 3 be an integer,for fixed integer m with m≥ 1,we denote Z_q*the least nonnegative reduced residue modulo q,let（am）g≡am（mod q）where 1 ≤（a^m）_q≤q,a∈q,a∈Z_q*,the paper study the maximal difference of powers of an element modulo q.2.Let q,h,k,l≥ 2 be integers,a be any positive integer with（a,q）=1,let 0<δ1,δ2≤1 be real number,the paper study asymptotic estimation of （?）Also,let integer q≥ 2,δ be real number with 0<δ<1,m≠be a positive integer,for an arbitrary set A（?）Z_q*,we can get （?）3.Let q,k>2 be positive integers,given k-dimension integer vectors a=（a₁,...a_k）,m=（m₁,...,m_k）,t=（t₁,...t_k）.Let a=（a₁,...,a_k）∈Z*qk be k-dimension integer vector,define（?）.where（ajmj）q denotes the least positive residue of ajmj modulo q,for j=1,...,k.Let the set（?）Z_t1 ×…×Z_tk,and define composite map（?）,which contained in the set（?）.Obtaining the asymptotic formula as follows （?）.