Balancing with binomial coefficients

• Published in: International journal of number theory Vol. 10; no. 7; pp. 1729 - 1742
• Main Authors: Komatsu, Takao, Szalay, László
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.11.2014; World Scientific Publishing Co. Pte., Ltd
• Subjects: Arrays; Binomial coefficients; Integers; Diophantine equations; binomial coefficients; balancing problems
• ISSN: 1793-0421; 1793-7310
• DOI: 10.1142/S1793042114500523

In this paper, we study a balancing problem of ordinary binomial coefficients. The diop-hantine equation $$ \left(\begin{array}{@{}c@{}} 0\\ k \end{array}\right) +\left(\begin{array}{@{}c@{}} 1\\ k \end{array}\right) +\cdots+\left(\begin{array}{@{}c@{}} x-1\\k \end{array}\right)=\left(\begin{array}{@{}c@{}} x+1\\ \ell \end{array}\right)+\cdots+\left(\begin{array}{@{}c@{}} y-1\\ \ell \end{array}\right)$$ in positive integers x and y ≥ x + 2 will be investigated, where k and ℓ are given positive integers. In case of k < ℓ the basic tool is the Runge method. If k ≥ ℓ, then the main results state effective and non-effective finiteness theorems on the subject matter. Further, computational results are also reported including the complete solutions when 1 ≤ k, ℓ ≤ 3, and a computer search otherwise.