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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
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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}

References [Enhancements On Off] (What's this?)

  • 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

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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