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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
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