From the Wolfram MathWorld page on squarefree numbers,
"There is no known polynomial time algorithm for recognizing squarefree integers or for computing the squarefree part of an integer."
Wolfram MathWorld says that
"This problem is an important unsolved problem in number theory because computing the ring of integers of an algebraic number field is reducible to computing the squarefree part of an integer."
Would there be any other applications for an algorithm that recognizes squarefree integers?