$N=512$ is an example for a positive integer $N$, such that the totient values of $N$ , $N+1$ , $N+2$ are all perfect squares.
Does a positive integer $N$ exist , such that the totient values of $N,N+1,N+2,N+3$ are all perfect squares ? In other words, can $4$ consecutive positive integers have square totient-values ?
Upto $N\le 10^8$, no such number exists.