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在数学解题过程中,直接举出满足条件的数学对象(或反倒),导致结论的肯定(或否定),或者利用具体问题的特殊性,设计一个框架,通过问题的转化来解决,这种解题方法称为构造法,构造法是一种重要的数学思想方法,应用构造法证明某些整除性问题,常可收到事半功倍的效果。常用的构造法有如下几种: 1 构造函数例1 证明7|sum from k=1 to 1986(2~k)(《数学通报》1986年6月号问题征解第416题) 证明构造函数 f(χ)=2(χ+1)~(662)-2, 显然,f(χ)是χ的整系数多项式。∵f(0)=0, ∴χ|f(χ),故7|f(7)。而f(7)=2·8~(662)-2=2(2~(1986)-1)=sum from k=1 to 1986 (2~k)得证。
In mathematics problem solving process, directly enumerate the mathematical objects (or conversely) satisfying the conditions, lead to the affirmation (or negation) of the conclusions, or use the specificity of specific problems to design a framework that can be solved through the transformation of the problems. The problem-solving method is called construction method, and the construction method is an important method of mathematics. Applying the construction method to prove certain divisibility problems can often receive a multiplier effect. Commonly used construction methods are as follows: 1 Constructor Example 1 Prove that 7|sum from k=1 to 1986 (2~k) (The “Junction Bulletin” June 1986 problem collection question 416) Prove the constructor f (χ) = 2(χ+1)~(662)-2. Obviously, f(χ) is an integral polynomial of χ. ∵f(0)=0, ∴χ|f(χ), so 7|f(7). And f(7)=2·8~(662)-2=2(2~(1986)-1)=sum from k=1 to 1986 (2~k).