Artin-type infinitude of lengths with universally invertible necklaces

Establish that, for every prime p, there are infinitely many lengths n that are not powers of p such that every aperiodic necklace of length n over the alphabet {0,1,...,p−1} with nonzero weight modulo p has an invertible Burrows–Wheeler matrix.

Background

The paper relates invertible necklaces to normal bases of finite fields and to p-rooted primes. It proves that if every aperiodic necklace of length n with nonzero weight modulo p is invertible, then n must be either a power of p or a p-rooted prime.

The authors then identify this property with a reformulation of Artin’s conjecture on the infinitude of p-rooted primes. The unresolved problem is therefore to prove the existence of infinitely many non-power-of-p lengths satisfying universal invertibility for aperiodic necklaces of nonzero weight.

References

There are infinitely many lengths $n$, different from a power of $p$, for which every aperiodic necklace of length $n$ over $\Sigma_p$ with non-zero weight modulo $p$ has invertible BWT matrix.

— Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform  (2502.12844 - Fici et al., 18 Feb 2025) in Conjecture immediately following Corollary in Section 4, “Invertible necklaces and Reutenauer groups”