engleski

A problem of Diophantus, Fermat and Euler and its generalizations

A set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements, increased by 1, is a perfect square. Diophantus himself found a set of four positive rationals with the above property, while the first Diophantine quadruple, the set {;1, 3, 8, 120};, was found by Fermat. In 1969, Baker and Davenport proved that this particular quadruple cannot be extended to a Diophantine quintuple. A folklore conjecture is that there does not exist a Diophantine quintuple. On the other hand, Euler was able to add the fifth positive rational, 777480/8288641, to the Fermat’ s set. In 2004, we have proved that there does not exist a Diophantine sextuple and there are only finitely many Diophantine quintuples. However, the bounds for the size of the elements of a (hypothetical) Diophantine quintuple are huge (largest element is less than 101026 ), so the remaining cases cannot be checked on a computer. The stronger version of this conjecture states that if {;a, b, c, d}; is a Diophantine quadruple and d > max{;a, b, c};, then d = a + b + c + 2abc + 2
(ab + 1)(ac + 1)(bc + 1). The problem of extension of given Diophantine triple to a quadruple leads to solving a system of Pellian equations, and this leads to finding the intersection of binary recursive sequences. Several variants of the congruence method are used to obtain lower bounds for solutions of the system of Pellian equations. In order to obtain upper bounds for solutions, we use recent results, due to Matveev and Mignotte, on linear forms in logarithms of three algebraic numbers.We will illustrate these methods and steps of the proof on the family of Diophantine triples {;k&#8722; 1, k+ 1, 16k3 &#8722; 4k};. In the recent joint work with Yann Bugeaud and Maurice Mignotte we were able to prove that for this family the strong conjecture of unique extension to a Diophantine quadruple is valid. Concerning rational Diophantine m-tuples, it is expected that there exist an absolute upper bound for their size. Such a result will follow from the Lang conjecture on varieties of general type. Related problem is to find an upper bound Mn for the size of D(n)-tuples (for given non-zero integer n), i.e. sets of positive integers with property that xy + n is perfect square for all of its distinct elements x, y. Again, the Lang conjecture implies that there exist an absolute upper bound for Mn (independent on n). However, at present, the best known upper bounds [Dujella, 2004] are of the shape Mn < clog |n|. Recently, in our joint paper with Florian Luca, we were able to obtain an absolute upper bound for Mp, where p is a prime. Let us mention that several examples of rational Diophantine sextuples are known [Gibbs, 1999-2006], but it is not know whether there exist infinitely many such sextuples. An interesting open question arises even when we consider the existence of quadruples. Namely, we stated the following conjecture: if n is not a perfect square, then there exist only finitely many D(n)-quadruples. It is very easy to verify the conjecture in case n &#8801; 2 (mod 4) (then there does not exist a D(n)-quadruple [Brown, 1984]), and recently, in the joint work with Clemens Fuchs and Alan Filipin, we have proved this conjecture in cases n = &#8722; 1 and n = &#8722; 4.

Diophantine equations; Pellian equations; elliptic curves

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