All congruent numbers less than 40000 (Q1130272)

The author finds the complete list of all rational integers \(0<n<42533\) such that the elliptic curve \(E_n\) with a model \(y^2=x^3-n^2x\) has rank over \({\mathbb Q}\) greater than or equal to \(1\). This condition is also equivalent to the one of \(n\) to be a congruent number. This text improves a former computation done by \textit{H. Wada} and \textit{M. Tairo} [Proc. Japan Acad., Ser. A 70, 154-157 (1994; Zbl 0820.14040)]. The author uses \(2\)-descent to reduce the problem to the solvability of certain diophantine equations, following, for example, \textit{J. H. Silverman} and \textit{J. Tate} in [Rational points on elliptic curves, Springer (1992; Zbl 0752.14034)]. Other problems are discussed concerning the elliptic curves \(E_n\), such as to precisely estimate their rank or to explicitly compute some quotients of their Shafarevich-Tate groups.