An extension of Lucas’ theorem
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- by Hong Hu and Zhi-Wei Sun PDF
- Proc. Amer. Math. Soc. 129 (2001), 3471-3478 Request permission
Abstract:
Let $p$ be a prime. A famous theorem of Lucas states that $\binom {mp+s}{np+t}\equiv \binom mn\binom st\ (\operatorname {mod} p)$ if $m,n,s,t$ are nonnegative integers with $s,t<p$. 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 $0$ and $1$.References
- Leonard Eugene Dickson, History of the theory of numbers. Vol. I: Divisibility and primality. , Chelsea Publishing Co., New York, 1966. MR 0245499
- Robert D. Fray, Congruence properties of ordinary and $q$-binomial coefficients, Duke Math. J. 34 (1967), 467–480. MR 213287
- Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
- William A. Kimball and William A. Webb, Some congruences for generalized binomial coefficients, Rocky Mountain J. Math. 25 (1995), no. 3, 1079–1085. MR 1357110, DOI 10.1216/rmjm/1181072205
- Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math. 396 (1989), 212–219. MR 988552, DOI 10.1515/crll.1989.396.212
- A. Schinzel, Primitive divisors of the expression $A^{n}-B^{n}$ in algebraic number fields, J. Reine Angew. Math. 268(269) (1974), 27–33. MR 344221, DOI 10.1515/crll.1974.268-269.27
- C. L. Stewart, Primitive divisors of Lucas and Lehmer numbers, Transcendence theory: advances and applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976) Academic Press, London, 1977, pp. 79–92. MR 0476628
- Zhi Wei Sun, Reduction of unknowns in Diophantine representations, Sci. China Ser. A 35 (1992), no. 3, 257–269. MR 1183711
- Paul M. Voutier, Primitive divisors of Lucas and Lehmer sequences, Math. Comp. 64 (1995), no. 210, 869–888. MR 1284673, DOI 10.1090/S0025-5718-1995-1284673-6
- Brad Wilson, The Fibonacci triangle modulo $p$, Fibonacci Quart. 36 (1998), no. 3, 194–203. MR 1627408
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
- MR Author ID: 254588
- Email: zwsun@nju.edu.cn
- 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
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1860478