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A simple proof of a curious congruence by Sun

Author(s): Zun Shan; Edward T. H. Wang
Journal: Proc. Amer. Math. Soc. 127 (1999), 1289-1291.
MSC (1991): Primary 11A07, 11A41
Posted: January 27, 1999
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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:

$\begin{displaymath}\sum^{(p-1)/2}_{k=1}\frac{1}{k\cdot 2^k}\equiv\sum^{[3p/4]}_{k=1} \frac{(-1)^{k-1}}{k}\quad(\mathrm{mod}\,p).\end{displaymath}$

References:

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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