Diophantine Equations Of Third-order Recursive Sequences

Diophantine equation of third-order recursive sequence is an important content and research subject in number theory.In this paper,we mainly study the distribution of factors containing 3 in Tribonacci sequence,and find all the Tribonacci number and Narayana’s cows number that can be expressed as factorial,double factorial and factorial product by means of distribution method.In addition,Baker’s method is used to study the relationship between the product of Narayana’s cows numer and repdigit,b-repdigit,and some theoretical results are obtained.This paper is divided into five chapters:The first chapter introduces the background and research status of recursive sequences,and analyzes the application of Diophantine equations related to factorial,repdigit and b-repdigit in Tribonacci sequence and Narayana’s cows sequence.The second chapter gives some basic knowledge of this paper.In chapter 3,the 3-adic valuation of Tribonacci sequence is given.With the help of the 3-adic valuation,it is shown that only T2,T3,and T4 in the sequence can be expressed as double factorials.Only T5 can be expressed as the product of two factorials.Only T5 and T8 can be expressed as the product of two double factorials.That is,Diophantine equation Tn=m!! has only solutions(n,m)∈ {(2,1),(3,1),(4,2)}.Tn=m1!m2! has only solution(n,m1,m2)=(5,2,2)and Tn=m1!!m2!!has only solutions(n,m1,m2)∈{(5,2,2),(8,3,4),(8,4,3)}.In chapter 4,with the help of the 3-adic valuation of Narayana’s cows sequence,it is shown that only N1,N2,N3,and N4 in the sequence can be expressed as double factorials.Only N2,N3,N4,and N7 can be expressed as the product of two double factorials.In addition,we also studied the relationship between the product of consecutive Narayana’s cows sequences and repdigits,and proved the Diophantine equation Nn…Nn+(l-1)=a((10m-1)/9)has solution N13=88.And then based on the representation of consecutive Narayana’s cows sequences as repdigits,we continue to generalize and study the relationship between the product of consecutive Narayana’s cows sequences and b-repdigits,and prove the Diophantine equation Nn…Nn+(l-1)=a((9m-1)/8).The fifth chapter summarizes the main work of this paper and looks into the future research work.