On Convergents of Proper Continued Fractions

The paper’s Theorem 3.11 states exactly the sought equivalence for even candidates: p = p2, q = q2 occurs iff there exists a divisor a | p with Tp(x) + Ta(x) > 1; it proves this via a Beatty-sequence characterization and the identity q = ⌊p/x⌋ + 1 for even candidates . The candidate model independently proves the same equivalence by an explicit tail-parameterization t of the PCF, deriving formulas for T_{p2}(x) and T_{a1}(x) that force T_{p2}(x) + T_{a1}(x) > 1 in one direction and construct digits from a|p with t = (1−Tp)/Ta in the other. The two arguments are correct and substantively different in method: the paper uses Beatty sequences, while the model uses explicit algebra on the first two digits and tail. Key identities p2 = a1 b2 and q2 = b1 b2 + a2 used in both align with equation (10) in the paper , and the uniqueness of the even candidate denominator q = ⌊p/x⌋ + 1 matches Lemma 3.3 . The fractional-part map notation TN(x) = {N/x} is consistent across both .