While skimming the book *Concrete Mathematics*, (edit: first edition) I came across the following problem, which is listed there as a Research Problem: (Chapter 5, Exercise 96)

Is ${2n \choose n}$ divisible by the square of a prime for all $n > 4$.

This problem looked to me much simpler than a divisibility problem that I found on MO (look here), but then again, I guess in number theory, the simpler the problems looks, the harder it usually is!

The nice form of this problem has made me very curious to find out more about it. But because I do not have more than a fleeting acquaintance with number theory, I don't know what search keywords would be useful to gain more information about this problem.

Thus, could somebody please tell me more about this problem and its current status?