  The class of $N$-Koszul graded algebras of finite global dimension has gained
lots of attention in recent years, especially in the study of Artin-Schelter
regular algebras. While structurally rich and concrete, the only known examples
of such algebras are either when $N = 2$, i.e. the algebra is Koszul, or when
$N = 3$. Under a mild Hilbert series assumption, we rule out the existence of
$N$-Koszul graded algebras of finite global dimension for $N$ not prime.
Furthermore, we establish strong restrictions on the global dimension of such
algebras. This suggests that perhaps the existence of 3-Koszul algebras with
finite global dimension and `nice' Hilbert series is an anomaly.
