Integer Solution for a Class of Indeterminate Equations

The mathematical discipline of number theory has a long history. It originated in the ancient Greek period in natural mathematics and geometry. With the progress of science and technology, people began to recognize the importance of “number,” which led to the rapid development of number theory and gradually becoming an independent discipline. Indeterminate equations are one of the most important branches of number theory and have always received a lot of attention from the mathematical community. The problem of integer solutions of indeterminate equations was also called the Diophantus equation because of the profound study of these equations by Diophantus in the early third century. However, there is no unified method for solving the Diophantus equation today, so the investigation of its integer solution often needs to be combined with previous studies. This paper focuses on the application of Gaussian rings of integers and congruence theory to the study of integer solution problems for indeterminate equations of the form x^2+4^k=4y^n (k=1,2 n=3,5..). The study generalizes the understanding of the local distribution rules in specific situations, which helps to simplify the discussion of the classification of this category of problems and provides alternative ideas for subsequent more in-depth studies by others. This paper concludes that every such indeterminate equation can be discussed in x ≡ 1 (mod 2) and x ≡ 0 (mod 2). A template for discussion is provided for the first scenario, and only when 2k-1=n can there exist only an integer solution (x,y)=(±2^k,3) for the second scenario.