engleski

On a problem of Diophantus and Euler

Diophantus studied the problem of finding numbers such that the product of any two of them increased by the sum of these two gives a square. He found two triples {;4, 9, 28}; and {;3/10, 7/10, 21/5}; satisfying this property. Euler found a quadruple {;5/2, 9/56, 9/224, 65/224}; and asked if there is an integer solution of this problem. In this talk we will describe a construction of an infinite family of rational quintuples with the same property. The construction is based on the fact that there are infinitely many rational points on the curve y^2 = -(x^2-x-3)(x^2+2x-12). We will also present a recent joint result with Clemens Fuchs, where we proved that there does not exist a set of four positive integers with the above property.

Diophantine equations; elliptic curves; rational points

nije evidentirano