Rule out Pisot-degree-3 powers to establish transcendence of Mills' constant

Show that there does not exist a positive integer m such that Mills' constant ξ_3, defined as the smallest real number ξ > 1 with floor(ξ^{3^k}) prime for every k ∈ N, satisfies that ξ_3^{3^m} is a Pisot number of degree 3. Establishing this would complete the proof that ξ_3 is transcendental.

Background

The paper proves that for base exponent c ≥ 4 the smallest prime-representing constant ξ_c is transcendental. For c = 3 (Mills’ constant), the main result shows a dichotomy: either ξ_3 is transcendental, or there exists m ∈ N such that ξ_3^{3^m} is a Pisot number of degree 3.

To conclude transcendence in the c = 3 case, it is therefore sufficient to exclude the latter possibility. The author explains the difficulty of this step in Remark~\ref{Remark-b=3}, where attempting to control the conjugate contributions leads to terms b_k and e_k in the cubic recurrence p_{k+1} = p_k^3 − 3 b_k p_k + 3 e_k that resist the methods used for degrees 2 and for c ≥ 4.