We consider , the number of solutions to the equation in nonnegative integers and integers , for given integers , , , and . When , we show that except for a finite number of cases all of which satisfy for each solution; when , we show that except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving solutions.
The equation , to be solved in non-negative rational integers , has been mentioned by Masser as an example for which there is still no algorithm to solve completely. Despite this, we find here all the solutions. The equation , to be solved in non-negative rational integers and a rational integer , has been mentioned by Corvaja and Zannier as an example for which the number of solutions is not yet known even to be finite. But we find here all the solutions too; there are in fact only six.
In this paper, we discuss variations on the Brocard-Ramanujan Diophantine equation.