Solving the Inherited Perfect Square Problem

For a long time, people have been baffled by trying to solve the “Perfect Square Problem . From it, more problems are inherited. The inherited perfect square problem in this paper is one example. It is stated as follows: Given any integer N, try to find positive integer A and B, such that both A2 + B2 and A2 + (B - NA) 2 are perfect squares. In this paper, two different approaches for solving the problem are introduced. The first one is the Forward-Algorithm, which takes advantage of logical common thinking and modifes Dahlquist\u27s algorithm to demonstrate A and B. Hence, the Forward-Algorithm is easy to understand. By running its program, all solutions, if one exists, can be obtained, including the smallest one. The second approach is the Backward-Algorithm, which is based on the method of infinite descent and a revision of Dahlquist\u27s algorithm. Hence, there is a great deal of mathematics involved. The Backward-Algorithm will not guarantee that we find the smallest solution or all solutions when a solution does exist. The time complexity of the Forward-Algorithm and the Backward-Algorithm is also discussed. The Backward-Algorithm is strongly preferred from the point of time complexity