engleski

On the extendibility of Diophantine pairs

A set of m positive integers is called a Diophantine m-tuple if the product of any two of its distinct elements increased by 1 is a prefect square. There is a folklore conjecture that there does not exist a Diophantine quintuple. Moreover, there is a stronger version of that conjecture, that every Diophantine triple can be extended to a quadruple with a larger element in the unique way. Precisely, if {; ; a, b, c, d}; ; is Diophantine quadruple such that a < b < c < d, then d=d_{; ; +}; ; =a+b+c+2(abc+rst), where r, s and t are positive integers satisfying r^2=ab+1, s^2=ac+1 and t^2=bc+1. Let {; ; a, b, c, d}; ; such that a < b < c < d be a Diophantine quadruple. In this talk we give an upper bound for minimal c such that d ≠ d_{; ; +}; ; . It helps us to prove the strong version of the conjecture for various families of Diophantine triples. As corollary it furthermore implies the non-extendibitily of parametric families of Diophantine pairs to a quintuple.

Diophantine m-tuples

nije evidentirano