Open Access
October 2015 A note on balancing binomial coefficients
Shane Chern
Proc. Japan Acad. Ser. A Math. Sci. 91(8): 110-111 (October 2015). DOI: 10.3792/pjaa.91.110


In 2014, T. Komatsu and L. Szalay studied the balancing binomial coefficients. In this paper, we focus on the following Diophantine equation \begin{equation*} \binom{1}{5}+\binom{2}{5}+…+\binom{x-1}{5}=\binom{x+1}{5}+…+\binom{y}{5} \end{equation*} where $y>x>5$ are integer unknowns. We prove that the only integral solution is $(x,y)=(14,15)$. Our method is mainly based on the linear form in elliptic logarithms.


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Shane Chern. "A note on balancing binomial coefficients." Proc. Japan Acad. Ser. A Math. Sci. 91 (8) 110 - 111, October 2015.


Published: October 2015
First available in Project Euclid: 5 October 2015

zbMATH: 06616665
MathSciNet: MR3403941
Digital Object Identifier: 10.3792/pjaa.91.110

Primary: 11D25 , 11G05 , 11Y50

Keywords: Balancing problem , binomial coefficient , linear form in elliptic logarithms

Rights: Copyright © 2015 The Japan Academy

Vol.91 • No. 8 • October 2015
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