Find a positive integer $n$ such that $n/2$ is a square, $n/3$ is a cube and $n/5$ is a fifth power.
Above problem was given in the text “An Introduction to the Theory of Numbers” by “Ivan Niven and Zuckerman".
I tried to generalize the problem statement as:
Find a positive integer $n$ such that $n/2$ is a square, $n/3$ is a cube, $n/5$ is a fifth power, $n/7$ is a seventh power ... $n/p_{k}$ is a $p_{k}$-th power.
I have found the solution for this and trying to generalize it more. Now, How can I know that this work is authentic i.e. it has not been done by anyone earlier? Any sort of help is welcomed.