On a diophantine problem of P. Erdős (Q689677)

By using elementary algebraic number theory, L. J. Mordell solved the equation \(k^ 3 - k = \ell^ 2 - \ell\) in integers \(k\), \(l\) [Pac. J. Math. 13, 1347-1351 (1966; Zbl 0124.274)]. Recently \textit{J. H. E. Cohn} determined all the integer solutions of the equation \(k^ 3 + k = \ell^ 2 - \ell\) [Math. Scand. 68, 171-179 (1993; Zbl 0755.11009)]. It has been asked by Erdős to determine all the integer solutions of the equation \(k^ 3 - k^ 2 = \ell^ 2 - \ell\). In this note, following the arguments of Mordell, the author shows that the only integer solutions of the above equation are given by \((k,\ell) = (0,0),(1,0),(0,1)\) and \((1,1)\).