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

Bonciocat, Nicolae Ciprian; Cipu, Mihai; Mignotte, Maurice

Publ. Mat., Barc. 56, No. 2, 279-304 (2012).

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.

MSC:

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

Keywords:

Diophantine \(m\)-tuples; Pell equations; linear forms in logarithms

Software:

PARI/GP

References:

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