An extension of Lucas' theorem

Authors:
Hong Hu and Zhi-Wei Sun

Journal:
Proc. Amer. Math. Soc. **129** (2001), 3471-3478

MSC (2000):
Primary 11B39; Secondary 11A07, 11B65

DOI:
https://doi.org/10.1090/S0002-9939-01-06234-7

Published electronically:
June 8, 2001

MathSciNet review:
1860478

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be a prime. A famous theorem of Lucas states that if are nonnegative integers with . In this paper we aim to prove a similar result for generalized binomial coefficients defined in terms of second order recurrent sequences with initial values and .

**1.**L. E. Dickson,*History of the Theory of Numbers*, vol. I, Chelsea, New York, 1952, p. 396. MR**39:6807a****2.**R. D. Fray,*Congruence properties of ordinary and -binomial coefficients*, Duke Math. J.**34**(1967), 467-480. MR**35:4151****3.**K. Ireland and M. Rosen,*A Classical Introduction to Modern Number Theory (Graduate texts in mathematics; 84), 2nd ed.*, Springer-Verlag, New York, 1990, pp. 191-192. MR**92e:11001****4.**W. A. Kimball and W. A. Webb,*Some congruences for generalized binomial coefficients*, Rocky Mountain J. Math.**25**(1995), 1079-1085. MR**96i:11004****5.**D. E. Knuth and H. S. Wilf,*The power of a prime that divides a generalized binomial coefficient*, J. Reine Angew. Math.**396**(1989), 212-219. MR**90d:11029****6.**A. Schinzel,*Primitive divisors of the expression in algebraic number fields*, J. Reine Angew. Math.**268/269**(1974), 27-33. MR**49:8961****7.**C. L. Stewart,*Primitive divisors of Lucas and Lehmer sequences, in: Transcendence Theory: Advances and Applications (A. Baker and D.W. Masser, eds.), Academic Press, New York*, 1977, pp. 79-92. MR**57:16187****8.**Zhi-Wei Sun,*Reduction of unknowns in Diophantine representations*, Scientia Sinica (Ser. A)**35**(3) (1992), 257-269. MR**93h:11039****9.**P. M. Voutier,*Primitive divisors of Lucas and Lehmer sequences*, Math. Comp.**64**(1995), 869-888. MR**95f:11022****10.**B. Wilson,*Fibonacci triangles modulo*, Fibonacci Quart.**36**(1998), 194-203. MR**99d:11014**

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Additional Information

**Hong Hu**

Affiliation:
Department of Mathematics, Huaiyin Normal College, Huaiyin 223001, Jiangsu Province, People’s Republic of China

**Zhi-Wei Sun**

Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

Email:
zwsun@nju.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-01-06234-7

Received by editor(s):
April 18, 2000

Published electronically:
June 8, 2001

Additional Notes:
The second author is responsible for all the communications, and supported by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, and the National Natural Science Foundation of P. R. China.

Communicated by:
David E. Rohrlich

Article copyright:
© Copyright 2001
American Mathematical Society