The study of the power mean of the generalized polynomial exponential sums plays a very important role in analytic number theory, and many classical number theory problems are closely related to it. In this article, we use the elementary methods and the properties of the exponential sums to study the calculating problem of one kind of fourth power mean of some special generalized polynomial exponential sums, and we give some exact calculating formulae for them.

The main purpose of this paper is using the elementary methods and the number of the solutions of some congruence equations to study the calculating problem of the fourth power mean of one special two-term exponential sums, and give an exact calculating formula for it.

The research on exponential sums is very important in number theory and practical application. This paper mainly discusses the properties of generalized two-term exponential sums. By means of elementary method, analytical method and the number of solutions of congruence equation, this paper studies the calculation of the fourth power mean of a kind of generalized two-term exponential sums, and gives an exact calculation formula and asymptotic formula. At the same time, the research results of this paper also provide some new and effective ideas and methods for the calculation of high-th two-term exponential sums, which is innovative.