On the diophantine equations \(ax^ 2 + bx + c = c_ 0c_ 1^{y_ 1} \cdots c_ r^{y_ r}\) (Q1919697)

A general algorithm is devised to determine all solutions to any diophantine equation of the type described in the title, where the coefficients are integers. As an application, all solutions of \[ x^2+1= 2^{y_1}5^{y_2}\quad\text{and} \quad x^2+x+1= 3^{y_1}7^{y_2}13^{y_3} \] are determined. These are important for group theoretical and geometric problems. Also solved is Ramanujan's diophantine equation \[ x^2+7= 2^y, \] which has connections to coding theory. For a more complete bibliographical reference to the previous equation see \textit{E. L. Cohen} [Number theory, Proc. 1st Conf. Can. Number Theory Assoc. 1988, 81-92 (1990; Zbl 0695.10015)].