Does frequency need to be an integer in order for a signal to be periodic?
It's sort-of a semantic issue, while:

$$ x[n] = \cos(2 \pi 3 n) $$

is clearly periodic with period of 1, the other discrete-time signal

$$ y[n] = \cos(2 \pi \sqrt{3} n) $$

is not periodic.   There is no integer \$N\$ such that \$y[n+N]=y[n] \quad \forall n \$.  While \$y[n]\$ is not periodic, the continuous-time counterpart to it 

$$ y(t) = \cos(2 \pi \sqrt{3} t) $$

is periodic because there is a real value \$T\$ such that \$y(t+T)=y(t) \quad \forall t \$.