Arithmetic Properties of Representation Dimensions in Grand Unified Theories

I examine the prime factorizations of the fundamental and adjoint representation dimensions of the three standard grand unified theory (GUT) gauge algebras: su(5), so(10), and e₆. All three carry prime factors exceeding 3 in at least one of these two dimensions. In contrast, the Standard Model gauge algebras su(2) and su(3) have both dimensions 3-smooth, meaning that every prime factor is at most 3. Among all simple compact Lie algebras, only four have this property: su(2), su(3), so(9), and sp(8). This classification was proved in [1] using Størmer's theorem and Zsygmondy's theorem. I show here that the Standard Model anomaly cancellation conditions force the hypercharge ratio structure 1 : 4 : −2 : −3 : −6, whose coefficients are all 3-smooth, and that this requires the number of colors Nc to satisfy Nc ≤ 3. The same bound follows independently from requiring Nc and Nc² − 1 to both be 3-smooth. No theoretical significance is claimed. I report these arithmetic facts and note which experiments would make them physically relevant.