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The problem of the extension of a parametric family of diophantine triples. (English) Zbl 0903.11010

A set of positive integers \(\{a_1,a_2,\ldots,a_m\}\) is said to have the property of Diophantus if \(a_ia_j+1\) is a perfect square for all \(1\leq i<j\leq m\). Such a set is called a diophantine \(m\)-tuple. There is a well-known diophantine quadruple for all integers \(k\geq 2\): the set \(\{k-1,k+1,4k,16k^3-4k\}\).
The author proves the following statement. Theorem. Let \(k\geq 2\) be an integer. If the set \(\{k-1,k+1,4k,d\}\) has the property of Diophantus, then \(d\) has to be \(16k^3-4k\). It is worth mentioning that this result was known only in the cases \(k=2\) and \(k=3\). To prove the mentioned theorem, the author applies a system of Pellian equations, a result of Rickert on simultaneous rational approximations, linear forms of logarithms and the Grinstead method.

MSC:

11D09 Quadratic and bilinear Diophantine equations
11D25 Cubic and quartic Diophantine equations
11J13 Simultaneous homogeneous approximation, linear forms
11J86 Linear forms in logarithms; Baker’s method
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