CG22 Conjecture A.10 on parity of N_k(n)

Prove that for odd integers k and integers n satisfying gcd(n,k)=gcd(n+1,k)=1, the quantity N_k(n), defined as the number of pairs of positive integers (b_1,b_2) with b_1,b_2 ≤ (k−1)/2, b_1+b_2 ≥ (k+1)/2, and b_2 ≡ n b_1 (mod k), satisfies N_k(n) ≡ ⌊(k+1)/4⌋ (mod 2).

Background

This conjecture originates in Chen–Gendron (CG22, Conjecture A.10) and arises in the determination of spin parity for odd k in genus zero and one. The conjecture connects spin parity questions to a parity count for solutions of a modular congruence with bounds.

The present paper proves this conjecture by reformulating the parity condition in terms of Jacobi symbols and reducing it to a combinatorial identity and standard facts about Jacobi symbols.

References

Conjecture [{Conjecture A.10]. For odd $k$ and $\gcd (n, k) = \gcd (n+1, k) = 1$, let $N_k(n)$ be the number of pairs of positive integers $(b_1, b_2)$ such that $b_1, b_2 \leq (k-1)/2$, $b_1 + b_2 \geq (k+1)/2$, and $b_2 \equiv n b_1 \pmod{k}$. Then we have\n N_k(n) \equiv \left\lfloor\frac{k+1}{4}\right\rfloor \pmod{2}.

— Parity of $k$-differentials in genus zero and one  (2602.03722 - Chen et al., 3 Feb 2026) in Conjecture (labelled Conjecture~\ref{conj}), Section 1 (Introduction and Statement of Results)