The problem/solution of counting the number of (primitive) necklaces (Lyndon words) is very well known.
But what about results giving sufficient conditions for a given necklace be primitive? For example, in the binary case, a necklace of length $N$ (00..00100..01) will be primitive whenever the number $M$ of zeros between the two 1's is such that $\gcd(N,M)=1$.
Any idea/references for additional results of this type?