斐波那契数列的倒数和与几个不定方程的求解

发布时间：2018-03-16 01:15

  本文选题：Pell方程　切入点：斐波那契数列　出处：《延安大学》2017年硕士论文　论文类型：学位论文

【摘要】：随着不定方程和斐波那契数列的不断发展,它的研究成果在数学以及其它领域都有着重要的作用,促使越来越多的学者对其进行更广泛深入地研究.本文从以下三方面的内容进行了研究:第一部分:利用初等方法研究了斐波那契数列3k子列的有限和,并证明了斐波那契数列的一个等式.第二部分:运用同余、Legendre符号、递归序列等性质,讨论了几个Pell方程的整数解的问题.1.给出了qx~2-(qn±5)y~2=±1(q≡±1,±3(mod10)是素数)型和ax~2—mqy~2=±1(m∈Z~+,5|a,q = ±1,±3(mod10)是素数,a是合数,amq是非完全平方数)型的Pell方程的几个结果.2.证明了当D = 2~n(n∈Z+)时,不定方程组x~2-12y~2=1与y~2-Dz~2=4只有平凡解(x,y,z)=(±7,±2,0).3.证明了当D=P_1 ···P_s(1≤s≤3),其中p_i(1≤s≤3)是互异的奇素数时,不定方程组x~2-30y~2 = 1 与y~2-Dz~2 = 4仅有正整数解 D = 483,(x,y)=(241,44,2).4.证明了当D=2p_1 ···p_s(1≤s≤4),其中pi(1≤s≤4)是互异的奇素数时,除开D = 2×3×337×673,不定方程组x~2-42y~2=1与y~2-Dz~2=4仅有平凡解(x,y,z)=(±13,±2,0).第三部分:运用同余、递归序列、Pell方程的解的性质,研究了形如x3±C = Dy~2的不定方程,最终证明方程x3±64 = 103y~2,x3-1 = 109y~2的整数解分别为:((x,y)=((?)4,0),(x,y)=(1,0).
[Abstract]:With the development of the indefinite equation and Fibonacci series, its research results play an important role in mathematics and other fields. In this paper, the following three aspects are studied: the first part: the finite sum of Fibonacci series 3k subseries is studied by using elementary method. And we prove an equation of Fibonacci sequence. The second part: the use of congruence Legendre symbol, recursive sequence and other properties, In this paper, the problem of integer solution of several Pell equations is discussed. Some results of Pell equation of type qx~2-(qn 卤5Q = 卤1Q 鈮，

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