Reciprocal Of The Fibonacci Numbers And The Solution Of Several Indeterminate Equations

With the continuous development of the indeterminate equation and Fibonacci numbers,their achievements play an important role both in mathematics and other fields and attract a growing number of scholars to study in depth.This thesis studies on the following three aspects:Firstly,by using the elementary methods,we study the finite sum of 3k sequence of Fibonacci numbers,and prove an equality of the Fibonacci numbers.Secondly,by using congruence,Legendre symbol,recurrence sequence and other properties,we discuss some problems of several Pell equations'integer solution.1.We obtain some conclusions about two kinds of Pell Equations of qx²-?qn±5?y² = ±1?q ? ±1,±3?mod10?is prime?and ax²-mqy² ?+1?m?Z+,5|a,q?±1,±3?mod 10?is a prime factor,amq is a non-square positive integer?.2.We prove that the system of indeterminate equations x²-12y² ?1 and y²-Dz² ?4 have only trivial solution?x,y,z?=?± 7,±2,0?when D?2n?n?Z+?.3.The system of indefinite equations x²-30y² = 1 and y²-Dz² = 4 has only posi-tive integer solution D = 483,?x,y,z?=?241,44,2?when D =pl··ps?1? s? 3??Ps?1?s?3?is diverse primes?.4.It is proved that?when D = 2p₁···p_s?1? s ? 4?,p,?1? s ?4?is diverse primes?,the system of indeterminate equations x²-42y² =1 and y²-Dz² = 4 has only trivial solution?x,y,z?=?± 13,±2,0?with the exceptions that D = 2 x 3 x 337 x 673.Thirdly,by using congruence,recurrence sequence and the properties of solution of Pell equation,we have studied a kind of indeterminate equation x3 ±C ?Dy² and proved respectively the equation x3 ±64 = 103y²,x3-1 = 109y² has only integer?x,y?=????4,0?and?x,y?=?1,0?.