一集合中任兩數乘積與完全平方數間關係之探討.pdfVIP

一集合中任兩數乘積 與完全平方數間關係之探討 李彥余 臺北市立建國高級 中學 Abstract The property of square numbers and how they consist is always one of the focal points in number theoritical researches. In this project, I will study the relation be- tween “the product of any two numbers that belong to a set of natural numbers” and “square numbers”. Surprisingly, I discover that the problem has a close relationship with Fibonacci sequence, which is often obeyed in nature. To be specific, the ultimate goal of this project is to find all of the sets of natural numbers p , p , , pn to 1 2 make p p k, 1 i j n and k , to be a square number (or to prove that it’s i j impossible to find the answer with the given n, k). The problem is too complicate for me to find all the solutions in a short time. Nevertheless, partial solutions under the situations below have been found so far: n, k 3, 1, 3, 1 or 4, 1. Besides, all the solutions of n, k 3, 1 lead to solutions of n, k 4, 1 and the minimum solutions for n 3. 摘摘摘要要要: 完全平方數之特性及其組成方式一直是數論的研究重點之一, 本研究計畫針對一 組正整數 中的任二數之乘積與完全平方數之關係進行一系列之研究. 令人驚奇的是, 我 發現此問題與 自然界 中常遵循的費 氏數列之規律有相當密切的關係. 具體而言, 本研究 的終極 目標為找 出所有的正整數集合 p , p , , pn 使其任二數 p 及 p 之乘積加 1 2 i j 上某一給定整數 k 均為一完全平方數, 亦即對所有的 1 i j n, p p k 均為一完 i j 全平方數並證明之 (或證明給定的 n, k 無法找到解答). 本 問題極為複雜, 我在短時間內並無法找 出所有解, 然本研究 目前 已找 出如下情況 的部分解答: