engleski

On a problem of Diophantus and Euler

Diophantus of Alexandria was interested in finding sets of (rational) numbers with the property that the product of any two of its distinct elements plus their sum is a perfect square. He gave the examples {; ; 4, 9, 28}; ; , {; ; 3/10, 21/5, 7/10}; ; . Euler gave an example of a set consisting of four rational numbers with this property, namely {; ; 5/2, 9/56, 9/224, 65/224}; ; . He asked the following: Euler's question: Is there a set consisting of four positive integers and the property that the product of any two distinct elements plus their sum is a perfect square? It is interesting to observe that xy + x + y = (x + 1)(y + 1) - 1 and therefore we may equivalently ask for sets of integers larger than 1 with the property that the product of any two distinct elements decreased by 1 is a perfect square. This is related to another problem in which already Diophantus was interested in. Recently jointly with A. Dujella we gave the answer to the question of Euler. Theorem (Dujella - Fuchs). The answer to Euler's question is no. More generally, we can consider the same problem in integers ; but if there is one negative element in D, then all have to be less then -1 and by changing all the signs simultaneously we get a set considered in the theorem. However, it is not natural to exclude zero from D and this leads to a much harder problem. If 0 in D, then all elements are squares. Recently, we were able to prove the following Theorem (Dujella - Filipin - Fuchs). There are at most finitely many sets D of four integers such that the product of any two distinct elements from this set plus their sum is a perfect square. Moreover, maxD < 10^10^23. The open problem remains to calculate all the remaining possibilities, the conjecture being that there is no such set at all.

simultaneous Pellian equations

Koautori: Andrej Dujella i Alan Filipin