关于Fibonacci三角形猜想k=11的证明  

A Proof of Fibonacci Triangles Conjecture with k=11

作　　者：林丽娟[1] 

机构地区：[1]重庆师范大学数学与计算机科学学院,重庆400047

出　　处：《重庆师范大学学报（自然科学版）》2008年第2期37-39,共3页Journal of Chongqing Normal University:Natural Science

基　　金：重庆市教委科研基金项目(No.KJ050807)

摘　　要：Fibonacci数列和Lucas数列的性质一直是数论中重要的研究内容之一,本文利用Fibonacci数列的性质研究了Fibonacci三角形猜想在k=11时的情形,讨论了以Fibonacci数Fn,Fn+11,Fn+11为边长并且面积为整数的三角形的存在性问题。首先假设猜想不成立,由边长和面积为整数,结合Fibonacci数列自身的性质得出边长之间所要满足的等量关系,然后对等式两边取模,利用Jacobi符号得出矛盾,从而证明了Fibonacci三角形猜想在k=11时成立,即不存在以Fibonacci数Fn,Fn+11,Fn+11为边长并且面积为整数的三角形。The Fibonacci sequence and the Lucas sequence has been one of the important research contents in number theory. In this paper, we consider Fibonacci triangles conjecture with k = 11 by the property of the Fibonacci sequence. A Fibonacci triangle is a triangle with integer area and sides whose lengths are Fibonacci numbers. As a first step, we suppose the conjecture is wrong, and then there exists triangle with integer area and sides whose lengths are Fibonacci numbers. We use the Jacobi symbol as a tool for proving that a number is not a perfect square. Finally we prove the nonexistence in Fibonacci triangles for k = 11 that there are no Fibonacci triangles with sides lengths Fn ,Fn＋11 ,Fn＋11.

关 键 词：FIBONACCI数 LUCAS数 FIBONACCI三角形 平方剩余 JACOBI符号 

分 类 号：O156[理学—数学]