engleski

The extension of some Diophantine triples

Diophantinem-tuple is the set of m positive integers such that the product of any two of them increased by 1 is a perfect square. One of the most interesting question, and problem various mathematicians tried to solve, is how large those sets can be. Recently, He, Togbe and Ziegler proved the folklore conjecture that there does not exist a Diophantine quintuple. There is a stronger version of that conjecture which states that every Diophantine triple can be extended to a quadruple with a larger element in a unique way. That is still an open problem.In this talk, we consider the extension of the infinite two-parameter family of Diophantine triples. More precisely, we prove the following: Let a and b be positive integers defined by a=KA^2, b=4KA^4+4εA with K, A positive integers and ε=±1. Define an integer c=c^{; ; ±}; ; _ν by c^{; ; ±}; ; _ν=1/(4ab)((√b±√a)^2(r+√ab)^{; ; 2ν}; ; +(√b∓√a)^2(r−√ab)^{; ; 2ν}; ; −2(a+b)), with ν positive integer. Then, there is a unique extension of the Diophantine triple {; ; a, b, c}; ; to a quadruple with a larger element. We also prove the stronger version of Diophantine quintuple conjecture for the triples of the form {;a, b, c};, where roughly a^2<b<4a^2.

Diophantine m-tuples

nije evidentirano