On \(D(-1)\)-quadruples. (English) Zbl 1354.11024

Summary: Quadruples \((a,b,c,d)\) of positive integers \(a<b<c<d\) with the property that the product of any two of them is one more than a perfect square are studied. Improved lower and upper bounds for the entries \(b\) and \(c\) are established. As an application of these results, a bound for the number of such quadruples is obtained.


11D09 Quadratic and bilinear Diophantine equations
11D45 Counting solutions of Diophantine equations
11B37 Recurrences
11J68 Approximation to algebraic numbers


Full Text: DOI Euclid


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