关于Fibonacci三角形猜想k=11的证明

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。

A Proof of Fibonacci Triangles Conjecture with k=11

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 lengthsFn,Fn 11,Fn 11.