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.


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