1) recurrent sequence
递归数列
1.
In this paper,the authors use recurrent sequence to prove that the diophantine equation x3-1=26y2 has only integer solutions(x,y)=(1,0),(3,±1),(313,±1086).
用递归数列方法证明了方程x3-1=26y2全部整数解是(1,0),(3,±1),(313,±1086)。
2.
By using congruence and recurrent sequence,the author has proved that the Diophantine equation x3+1=86y2 has only the integer solution(x,y)=(-1,0),(7,±2).
利用递归数列、同余式证明了丢番图方程x3+1=86y2仅有整数解(x,y)=(-1,0),(7,±2)。
3.
By using the method of recurrent sequence,the Diophantine equation x3-8=13y2 has been proven to have the only integer solution(x,y)=(5,±3) with gcd(x,y)=1.
利用同余式和递归数列的方法,证明了不定方程x3-8=13y2仅有适合(x,y)=1的整数解(x,y)=(5,±3)。
2) recursive sequence
递归数列
1.
In this paper,the author has proved that the Diophantine equation x3+64=21y2 has only an integer solution(x,y)=(-4,0),(5,±3) and then gives all integer solution of x3+64=21y2 by using the elementary methods such as recursive sequence,congruent fomula and quadratic residue.
利用递归数列、同余式和平方剩余几种初等方法,证明了不定方程x3+64=21y2仅有整数解(x,y)=(-4,0),(5,±3);给出了x3+64=21y2的全部整数解。
2.
In this paper,the author has proved that the Diophantine equation x3+27=7y2 has only an integer solution(x,y)=(-3,0),(1,±2) and then gives all integer solution of x3+27=7y2 by using the elementary methods such as recursive sequence,congruent fomula and quadratic residu
利用递归数列、同余式和平方剩余几种初等方法,证明了不定方程x3+27=7y2仅有整数解(x,y)=(-3,0),(1,±2);给出了x3+27=7y2的全部整数解。
3.
By applying the properties of determinant, the general formula of a usual type of linear recursive sequence is studied.
应用行列式的有关性质，研究了一般形式的线性递归数列的通项公式。
3) linear recursion sequence
线性递归数列
1.
Then, two conjectures are brought forward, which are proved that there are relations between the two deformations and a class of linear recursion sequences such as a_(n+k+l)=a_(n+k)+a_n.
从杨辉三角的两种基本变体即错位变体和克隆变体的概念,提出两个猜想,并证明两种变体的各行和与形如a_(n+k+l)=a_(n+k)+a_n的线性递归数列的对应关系,同时给出这类递归数列的两种通项公式1)。
4) Fractional Recurrent Sequence
分式递归数列
1.
The Period of a Class of Fractional Recurrent Sequence;
一类分式递归数列的周期性
5) recursive sequence
递归序列
1.
The relation between the iteration of projective function and the linear recursive sequences of order 2 is given.
先给出射影函数的迭代与 2阶线性递归序列的关系 ,进而得到此递归序列与Bernoulli数的一个恒等
6) recurrent sequence
递归序列
1.
This paper proves that the Diophantine equation has only integer solution with the help of the Pell method taking an integer>1 as module to make inconsistency,the natures of recurrent sequences and equivalent Pell equation.
采用对方程取某个正整数M>1为模来制造矛盾的同余法和利用递归序列的性质,以及Pell方程的性质,证明不定方程x3-1=13y2仅有整数解(x,y)=(1,0)。
2.
In this paper,the author has proved, with two method of contradictor recurrent sequences and congruence when modules of some positive integer M>1, that the Diophantine equation x~3+1=19y~2 has only integer solution(x,y)=(1,0).
利用两种初等的方法,即对方程取某个正整数M>1为模来制造矛盾的同余法和递归序列法,证明了不定方程x3 -1=19y2 仅有整数解(x,y)=(1,0),从而进一步的证明了方程x2 -19y2 =-13无整数解;方程x2 -3r2 =-3仅有整数解(1。
3.
With the method of recurrent sequence and congruences,proved that the Diophantine equation x3+1 =37y2has only integer solution(x,y)=(-1,0),(11,±6).
利用递归序列,同余式证明了丢番图方程x 3+1=37y2,仅有整数解(x,y)=(-1,0),(11,±6)。
补充资料：递归数列
| 递归数列 recursive sequence 一种用归纳方法给定的数列。例如，等比数列可以用归纳方法来定义，先定义第一项a1的值(a1≠0)，对于以后的项，用递推公式an+1＝qan（q≠0，n＝1，2，…）给出定义。一般地，递归数列的前k项a1，a2，…，ak为已知数，从第k＋1项起，由某一递推公式an+k＝f（an，an+1，…，an+k-1）( n＝1，2，…）所确定。k称为递归数列的阶数。例如 ，已知 a1＝1，a2＝1，其余各项由公式an+1＝an＋an-1（n＝2，3，…）给定的数列是二阶递归数列。这是斐波那契数列，各项依次为 1，1，2，3，5，8，13，21，…，同样，由递归式an+1－an ＝an－an-1( a1，a2为已知，n＝2，3，… ) 给定的数列，也是二阶递归数列，这是等差数列。 |
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