Consecutive binomial coefficients satisfying a quadratic relation. (English) Zbl 1121.11025

Luca, Florian; Szalay, László

Publ. Math. Debr. 69, No. 1-2, 185-194 (2006).

In the present paper the authors study the Diophantine equation \(A\binom{n}{k}^2+B\binom{n}{k+1}^2+C\binom{n}{k+2}^2=0\) with positive integers \((n,k)\), where \(A, B, C\) are fixed integers and \(\gcd(A,B,C)=1\), \(A>0\). They proved that the equation has at most finitely many effectively computable integer solutions \((n,k)\) with \(1\leq k <k+2\leq n-1\).

Reviewer: Sándor Turjányi (Debrecen)

MSC:

11D09

Quadratic and bilinear Diophantine equations

Keywords:

quadratic Diophantine equations; binomial coefficients