關于 Lucas 數列同余性質的研究

馬榮,張玉龍

(1.西北工業大學理學院,陜西 西安 710072;2.西安交通大學電信學院,陜西 西安 710049)

關于 Lucas 數列同余性質的研究

馬榮1,張玉龍2

(1.西北工業大學理學院,陜西 西安 710072;2.西安交通大學電信學院,陜西 西安 710049)

將二項式系數的性質應用到Lucas數列的研究中,并結合Fibonacci數列與Lucas數列的恒等式得到幾個有趣的Lucas數列的同余式.

Fibonacci數列;Lucas數列;恒等式;同余式

1 引言

當n是素數時,知道Ln模n同余于1,這一性質對部分n是合數的情況也成立.關于Fn與Ln的其它性質,請參見文獻[5-8].

2 幾個引理

為了證明定理,需要以下幾個引理.首先得到幾個關于Fn的新的恒等式.

引理2.1對任意非負整數m和n,有

3 定理的證明

[1] Paulo Ribenboim. The New Book of Prime Number Records[M]. New York: Springer, 1996.

[2] Franz Lemmermeyer. Reciprocity Laws[M]. New York: Springer, 2000.

[3] Derek Jennings. Some polynomial identities for the Fibonacci and Lucas numbers[J]. The Fibonacci Quar- terly, 1993,31(2):134-137.

[4] Constance Brown. Fibonacci Analysis[M]. Hoboken: Bloomberg Press, 2008.

[5] Zhang Wenpeng. Some identities involving the Fibonacci numbers and Lucas numbers[J]. The Fibonacci Quarterly, 2004,42:149-154.

[6] Ma Rong. Zhang Wenpeng. Several identities involving the Fibonacci numbers and Lucas numbers[J]. The Fibonacci Quarterly, 2007,45:164-170.

[7] Duncan R L. Applications of uniform distribution to the Fibonacci numbers[J]. The Fibonacci Quarterly, 1967,5:137-140.

[8] Kuipers L. Remark on a paper by R. L. Duncan concerning the uniform distribution mod 1 of the sequence of the Logarithms of the Fibonacci numbers[J]. The Fibonacci Quarterly, 1969,7:465-466.

[9] Apostol Tom M. Intruduction to Analytic Number Theorem[M]. New York: Springer-Verlag, 1976.

On the congruence properties of the Lucas numbers

Ma Rong1,Zhang Yulong2

(1.School of Science,Northwestern Polytechnical University,Xi′an 710072,China; 2.The School of Electronic and In formation Engineering,Xi′an Jiaotong University,Xi′an 710049,China)

In this paper, we have used the properties of the binomial coe±cient to study the Lucas sequences, after combining the binomial coe±cient with the identities involving Fibonacci sequences and Lucas sequences, we have got several interesting congruence identities involving the Lucas sequences.

Fibonacci numbers, Lucas numbers, identities, congruence formula

O156.1

A

1008-5513(2012)02-0269-06

2011-07-29.

國家自然科學基金(11071194);西北工業大學基礎研究基金(JC 201123).

馬榮(1982-),博士,講師,研究方向：初等數論與解析數論.

2010 MSC:11B39,11B83