本文是围绕一个方程,做为一个高三学生汇报自己如何读数学书籍的初步体会,敬请老师们指正。试证含有x,y的不定方程: x~2-2y~2=1有无穷多组(正)整数解。现把“格点和面积”书中证明过程摘录如下: “显然x~2-2y~2=1有解x=3,y=2, 即 (3+2 2~(1/2)(3-2 2~(1/2))=1。平方并化简,得(17+12 2~(1/2))(17-12 2~(1/2)=1, 即 17~2-2×12~2=1。即 x=17,y=12,是另一组解。取立方,四次方……,即得无穷多组解。”这个证明,实际上提供了不定方程x~2-2y~2=1的解法。一开始,感到这种解法非常巧妙。仿照这种方法,试解了方程x~2-2y~2=-1。显然,x=1,y=1,是这个方程的一组自然数解(以下“自然数解”均写“解”)。随后发现,必须将原方程两边立方,才能得到第二组解x=7,y=5。以后便是五次方,七次方…。这样,便初步掌握了这种类型的方程的解法。在翻阅一本名叫《趣味的数和图》时,其中第一章“趣味的数字”里有一题: This article is centered on an equation and serves as a preliminary experience for a senior high school student to report on how to read a math book. Please ask the teachers to correct me. Testify the indefinite equation with x,y: x~2-2y~2=1 has infinitely many sets of (positive) integer solutions. The excerpt of the proof process in the “Grid and Area” book is now extracted as follows: “Obviously x~2-2y~2=1 has solution x=3, y=2, ie (3+2 2~(1/2)(3). -2 2~(1/2)) = 1. Squaring and simplification results in (17+12 2~(1/2)) (17-12 2~(1/2)=1, ie 17~2 2×12~2=1. That is, x=17 and y=12. It is another set of solution. Take cubic, fourth power..., that is infinitely many sets of solutions.” This proof, in fact, provides the indefinite equation x The solution to ~2-2y~2=1.At first, I felt that the solution was very clever. Following this method, we solved the equation x~2-2y~2=-1. Obviously, x=1, y=1 , is a set of natural number solutions for this equation (the “natural number solutions” below all write “solutions”). It was later discovered that the original equation must be cubic on both sides to obtain the second set of solutions x=7, y=5. In this way, the solution to this type of equation is initially grasped. When you look through a book called “Fun Numbers and Figures,” the first chapter “Fun Numbers” contains a question: