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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
Published electronically:
January 27, 1999
MathSciNet review:
1486751
Full-text PDF Free Access
Abstract |
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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
(98e:11004), http://dx.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
(95f:11003), http://dx.doi.org/10.1090/S0002-9939-1995-1242105-X
- 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), 444-445. MR 98e:11004
- 4.
- Zhi-Wei Sun, A congruence for primes, Proc. Amer. Math. Soc. 123 (1995), 1341-1346. MR 95f:11003
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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:
http://dx.doi.org/10.1090/S0002-9939-99-04816-9
PII:
S 0002-9939(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
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