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

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 $$1\leq k <k+2\leq n-1$$.

### MSC:

 11D09 Quadratic and bilinear Diophantine equations