數(shù)論與代數(shù)實(shí)作

1、 數(shù)論與代數(shù)實(shí)作 左太政 /國立高雄師範(fàn)大學(xué)數(shù)學(xué)系 一、數(shù)論1.三角形數(shù)是形如的數(shù)，試證：必有無窮多個(gè)三角形數(shù)是整數(shù)的完全平方數(shù)。2.設(shè),試證：必有無窮多個(gè)使得不是質(zhì)數(shù)。3.n is perfect if the sum of the divisors of n is 2n. Suppose now n is an odd perfect number, show that n has at least 3 distinct prime factors.4.試求滿足的所有正整數(shù)。（Ans: =（2,2,2）5.試證：必有一個(gè)無窮正整數(shù)數(shù)列,使得 是一個(gè)整數(shù)的完全平方數(shù)，對(duì)於所有正整數(shù)均成立；並

2、求此數(shù)列的遞迴關(guān)係式。（Australian 2004,p.11, ）6.已知一數(shù)列滿足：試證：若為 的質(zhì)因數(shù)，則亦是 的質(zhì)因數(shù)。（Iran 2002-2003 p.2）7.Find all permutations of with the following property for every ：。（Iran 2002-2003 p.1）8.Find all real polynomials such that： implies that 。（Iran 2002-2003 p.3）9.試證：必不能表為三個(gè)正整數(shù)之完全平方數(shù)的和。並直接求出 的所有解。(初等數(shù)論: p.275)10.已知質(zhì)數(shù)

3、,試證：不定方程式有解的必要條件為必為 之形式。二、代數(shù)Theorem 1: （Power Mean Inequality）Let be positive real numbers and be real numbers such that . Then. Equality holds if and only if , where.Theorem 2: （Maclaurins Symmetric Mean Inequality）Let be positive real numbers. We define to be the average of all products of of the

4、s, i.e. , ,. Then . Equality holds if and only if .練習(xí)題:1.設(shè) 為正實(shí)數(shù)，試證：。2.設(shè)為正實(shí)數(shù)，如果,試證：.3.設(shè)為相異正整數(shù)，試證：.4.設(shè) 為正實(shí)數(shù)，試證：.5.已知正實(shí)數(shù) 滿足，試證：。6.已知正實(shí)數(shù) 滿足，試證：。7.已知一三角形之三邊長分別為,試證：。8.設(shè) 為正實(shí)數(shù)，試證：。9.設(shè) 為正實(shí)數(shù)，試證：。10.設(shè) 為正實(shí)數(shù)，試證：。11.Let be the lengths of the sides of a triangle. Prove that and determine when equality holds.12.Let be a triangle and an interior point of