Pisot power of Mills’ constant: existence of a degree‑3 case

Determine whether there exists an integer m≥1 such that ξ₃^{3^m} is a Pisot number of degree 3, where ξ₃ denotes the smallest real number ξ>1 such that ⌊ξ^{3^k}⌋ is a prime number for every positive integer k.

Background

The main theorem shows that either ξ₃ is transcendental or there exists m∈ℕ for which ξ₃^{3^m} is a Pisot number of degree 3. Determining which alternative holds is pivotal for settling the transcendence of Mills’ constant.

The authors highlight this precise structural dichotomy and isolate the remaining uncertainty as the existence (or nonexistence) of such a Pisot-power instance.