Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\alpha_{6,4}$ is not $6$-normal, and $\alpha_{b,c}$ is $b$-normal iff some prime of $c$ misses $b$

Mutual nondivisibility is not enough: $b = 6$, $c = 4$ is a counterexample, since every prime dividing 4 divides 6, and then the zero digit of $\alpha_{6,4}$ in base 6 has limiting frequency one. The exact criterion is that $\alpha_{b,c}$ is $b$-normal if and only if some prime dividing $c$ does not divide $b$. Sufficiency applies Bailey and Crandall's own exponential-sum estimate to the part of $c$ coprime to $b$; the repository's prior-art note records that Kano's 1993 theorem on Korobov-type constructions already reaches some non-coprime cases, $\alpha_{4,6}$ among them, so the classification rather than the method is what is new. Reversing the parameters, $\alpha_{4,6}$ is 4-normal, is also formalised.

Mutual nondivisibility is not enough: $b = 6$, $c = 4$ is a counterexample, since every prime dividing 4 divides 6, and then the zero digit of $\alpha_{6,4}$ in base 6 has limiting frequency one. The exact criterion is that $\alpha_{b,c}$ is $b$-normal if and only if some prime dividing $c$ does not divide $b$. Sufficiency applies Bailey and Crandall's own exponential-sum estimate to the part of $c$ coprime to $b$; the repository's prior-art note records that Kano's 1993 theorem on Korobov-type constructions already reaches some non-coprime cases, $\alpha_{4,6}$ among them, so the classification rather than the method is what is new. Reversing the parameters, $\alpha_{4,6}$ is 4-normal, is also formalised.