A simple proof of a curious congruence by Sun
Authors:
Zun Shan and Edward T. H. Wang
Journal:
Proc. Amer. Math. Soc. 127 (1999), 1289-1291
MSC (1991):
Primary 11A07, 11A41
DOI:
https://doi.org/10.1090/S0002-9939-99-04816-9
Published electronically:
January 27, 1999
MathSciNet review:
1486751
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Abstract | References | Similar Articles | Additional Information
Abstract: In this note, we give a simple and elementary proof of the following curious congruence which was established by Zhi-Wei Sun:
![\begin{displaymath}\sum^{(p-1)/2}_{k=1}\frac{1}{k\cdot 2^k}\equiv\sum^{[3p/4]}_{k=1} \frac{(-1)^{k-1}}{k}\quad(\mathrm{mod}\,p).\end{displaymath}](/proc/1999-127-05/S0002-9939-99-04816-9/gif-abstract/img1.gif)
- 1. Louis Comet, Advanced Combinatorics, D. Reidel Publishing Company, 1974.
- 2. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, Oxford, 1960.
- 3. Winfried Kohnen, A simple congruence modulo 𝑝, Amer. Math. Monthly 104 (1997), no. 5, 444–445. MR 1447978, https://doi.org/10.2307/2974738
- 4. Zhi Wei Sun, A congruence for primes, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1341–1346. MR 1242105, https://doi.org/10.1090/S0002-9939-1995-1242105-X
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Additional Information
Zun Shan
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu, 210097, People’s Republic of China
Edward T. H. Wang
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5
Email:
ewang@machl.wlu.ca
DOI:
https://doi.org/10.1090/S0002-9939-99-04816-9
Received by editor(s):
August 13, 1997
Published electronically:
January 27, 1999
Communicated by:
David Rohrlich
Article copyright:
© Copyright 1999
American Mathematical Society