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Diophantine m-tuples and 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. 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 10^10^26), 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+2sqrt{; ; (ab+1)(ac+1)(bc+1)}; ; . Diophantine quadruples of this form are called regular. In our future attempt to prove the Diophantine quintuple conjecture, our strategy will be to consider two different cases, depending on whether the hypothetical quintuple extends a regular or irregular Diophantine quadruple. 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. By a precise analysis of the initial terms of these sequences, we would like to improve inequalities which have to be satisfied by irregular quadruples. At present, it is known that d>c^3, and we intend to prove something like d>c^7. Such an inequality will allow us to apply very useful results (e.g. a result due to Bennett) on simultaneous Diophantine approximation of algebraic number close to 1. For the proof of extendibility of regular Diophantine quadruples, we intend to improve the congruence method, which is used to obtain lower bounds for solutions of the system of Pellian equations. We will again distinguish two cases depending of the form of the third element in the quadruple (one case is c=a+b+2sqrt{; ; ab+1}; ; ). In order to obtain upper bounds for solutions, we will use recent results, due to Matveev and Mignotte, on linear forms in logarithms of three algebraic numbers. The last step in the proof will certainly include an extensive verification on computers. Namely, we will have to check, by applying Baker-Davenport reduction method, that large (but hopefully reasonable) number of systems of Pellian equations have only trivial solutions. The proposed methods and steps of the proof have been recently tested, in the joint work with Yann Bugeaud and Maurice Mignotte, on the family of Diophantine triples {; ; k-1, k+1, 16k^3-4k}; ; , and 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. Very related problem is to find an upper bound M_n 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 M_n (independent on n). However, at present, the best known upper bounds [Dujella, 2004] are of the shape M_n < c log|n|. Recently, in our joint paper with Florian Luca, we were able to obtain an absolute upper bound for M_p, 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 == 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=-1 and n=-4.

Diophantine m-tuples

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