On the Representation of a Number as the Sum of Two Squares and a Prime

This paper shows that every integer and odd integer are a sum of a prime and two squares. We solved this by transforming the problem of the number of such representations into finding the number of integer solutions to the corresponding Diophantine equation. To find integer solutions to nonlinear or higher-order Diophantine equations is very challenging, whereas counting the number of integer solutions for the linear Diophantine equations is comparatively easier. The methods involve the combinatorial approximation method to obtain the number of integer solutions of the Diophantine equation with the countable solution sets. First, the number of integer solutions to the linear Diophantine equations with integer sets or sequences as solution sets can be determined by combinatorial methods. Second, the approximate number of integer solutions of the linear Diophantine equations with integer sets or sequences as solution sets is obtained using the approximate method. Finally, for cases where the solution sets and the number of solution sets of the variables in the linear Diophantine equations are either the same or different, we propose the approximate combinatorial method to compute the approximate number of integer solutions of the linear Diophantine equations whose solution sets are integer subsets or sequences of integer subsets. The results are consistent with the existing conclusions. The purpose of this paper is to verify the adaptability and correctness of the approximate combinatorial methods for solving the number of integer solutions of the general Diophantine equations with countable solution sets.