(In case of other

infinite groups this strategy is not well-defined.

For any positive integer k [greater than or equal to] 3, there exist

infinite group positive integers

One of the main ways to represent an

infinite group using only a finite amount of data is via a presentation.

Jozsef Sandor [3] proved that for any positive integer k [greater than or equal to] 2, there exist

infinite group positive integers ([m.sub.1], [m.sub.2], ...

The main purpose of this paper is using the elementary method and the Vinogradov's important work to prove the following conclusion: For any positive integer k [greater than or equal to] 3, there exist

infinite group positive integers ([m.sub.1], [m.sub.2], ..., [m.sub.k]) such that the equation Se ([m.sub.1] + [m.sub.2] + ...

The main purpose of this paper is using the elementary method to prove that for each k [greater than or equal to] 4, there exist

infinite group positive integers ([m.sub.1], [m.sub.2], ...

For example, Jozsef Sandor [4] proved that for any positive integer k [greater than or equal to] 2, there exist

infinite group positive integers ([m.sub.1], [m.sub.2], ...

In this paper, we using the elementary method to study this problem, and prove that for any integer n [greater than or equal to] 1, the inequality has

infinite group positive integer solutions ([x.sub.1], [x.sub.2], ..., [x.sub.n]).

has

infinite group positive integer solutions ([m.sub.1], [m.sub.2], ..., [m.sub.k]).

For any integer k [less than or equal] 2, we can find

infinite group numbers [m.sub.1],[m.sub.2], ..., [m.sub.k] such that:

For any integer k [greater than or equal to] 2, we can find

infinite group numbers [m.sub.1], [m.sub.2], ...