Balancing with binomial coefficients
Takao Komatsu, László Szalay · International Journal of Number Theory · 2014
In this paper, we study a balancing problem of ordinary binomial coefficients. The diop-hantine equation [Formula: see text] 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.