## 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