Tribonacci序列的丢番图方程  

Diophantine equations for Tribonacci sequences

作　　者：吉艳娇 杨鹏[1] JI Yanjiao;YANG Peng(School of Science,University of Science and Technology Liaoning,Anshan 114051,China)

机构地区：[1]辽宁科技大学理学院,辽宁鞍山114051

出　　处：《辽宁科技大学学报》2022年第4期294-299,共6页Journal of University of Science and Technology Liaoning

基　　金：辽宁省教育厅项目(2019LNJC08)。

摘　　要：给出了Tribonacci序列T_(n+2)=T_(n+1)+T_(n)+T_(n-1),T0=T_(1)=0,T_(2)=1中含有3的因子的分布,并以此证明该序列中仅有T_(2)、T_(3)、T_(4)可以表示成双阶乘;仅有T_(5)可以表示成两个阶乘的乘积;仅有T_(5)、T8可以表示成两个双阶乘的乘积,即:丢番图方程T_(n)=m!!仅有解(n,m)∈{(2,1),(3,1),(4,2)};T_(n)=m_(1)!m_(2)!仅有解(n,m_(1),m_(2))=(5,2,2);T_(n)=m_(1)!!m_(2)!!仅有解(n,m_(1),m_(2))∈{(5,2,2),(8,3,4),(8,4,3)}。In this paper,the(T_(n))n>0 is taken as the Tribonacci sequence defined by the recurrence T_(n+2)=T_(n+1)+T_(n)+T_(n-1) with T0=T_(1)=0,T_(2)=1,and the 3-adic valuation of T_(n) is given.With the 3-adic valuation,it is proved that only T_(2),T_(3),and T_(4) in the sequence can be expressed by double factorial,only T_(5) can be ex-pressed by the product of two factorials,and only T_(5) and T8 can be expressed by the product of two double factorials.That is,Diophantine equation T_(n)=m!!has only solutions of(n,m)∈{(2,1),(3,1),(4,2)},T_(n)=m_(1)!m_(2)!has only solution of(n,m_(1),m_(2))=(5,2,2),and T_(n)=m_(1)!!m_(2)!!has only solutions of(n,m_(1),m_(2))∈{(5,2,2),(8,3,4),(8,4,3)}.

关 键 词：Tribonacci序列 阶乘 双阶乘 丢番图方程 

分 类 号：O156.1[理学—数学]