This Paper is aimed to study a generalization of the Catlan Conjecture (The only nontrivial solution to& x^m-y^n=1& is x=3,m=2,y=2,n=3), that is, to determine the nontrivial solutions to& x^m-2y^n=1&. To solve this problem, I study some arithmetic properties of the solutions to Pell equation, or more precisely, when the minimum solution is fixed, the properties about factors of the recursive solutions to the equation. Furthermore, I apply these properties into the study of& x^m-2y^n=1& and obtain some successful results when n is restricted to be even. What’s more, some applications of these arithmetic properties in other problems are contained in the paper as well.

This research, initiated by a problem excerpted from Baidu, one of the most famous internet search engines in China, explores two types of movements, “converging curve” and “tracing curve” named by this research group, which really act in a regular pattern. In the research, by applying scores of mathematical methods, we found that the key to converging curves is to spot the invariant elements from the changeful motions. For tracing curves, we adopt estimating equations to get the solution. Due to the limit of time and the relevant knowledge of the research group, many unexpected problems constantly appeared which are quite far beyond our ability. Yet we still tried to apply some promising mathematical thoughts, and have achieved some exciting results. In the process of discovering and rediscovering, we did not succeed in finding a perfect solution to the problem. After all, as a group of middle school students fascinated by mathematics, creating and discovering more are the greatest delight of us all.

Our paper begins with a simple but beautiful subject given by Martin Gardner (Dividing ten gold coins and ten silver coins into two containers which look the same outside, then randomly choose one of the containers and take out one coin from the chosen container, Is the probability of getting golden coin becomes higher when the distributive method changes). And we change and consider the problem from some other points of view. 1. Researching the original subject’s probability. 2. Discussing the minimal unit of distributing golden coins. 3. Researching the number of containers. 4. Researching the minimal and maximal probability of getting golden coin which is the function expression of containers’ number. As a result, we will understand the subject more deeply. The essence of the subject tells us: the reason of probability’s changing can not only be interpreted by “Sample Space difference and Sample Point difference”, but also “Classical Probability’s infection to Geometry Probability”. According to this theory, we can model this subject: Gold→A and Silver→not A. Then we use the model in our example, successfully interpreting what J·BERTRAND had found. And we also get a conjecture and prove it. Further, we did some transformation of the original formulation according to the actual situation. This is our paper’s main content.