Exact Formulas for Coprime Representations of Even Integers Avoiding a Prime

1. Problem Statement

The paper addresses a classical problem in additive number theory: counting the number of ways an even integer $2n$ can be expressed as the sum of two positive integers, $h + k = 2n$ (with $h \le k$), subject to strict coprimality constraints.

Specifically, the author defines the counting function $g(2n, p)$ for a fixed prime $p \ge 5$:
$$g(2n, p) = \#\left\{ (h, k) \in \mathbb{Z}_{>0}^2 : h + k = 2n, \, h \le k, \, \gcd(h, 6p) = \gcd(k, 6p) = 1 \right\}$$
The condition $\gcd(x, 6p) = 1$ implies that the summands $h$ and $k$ must not be divisible by the primes 2, 3, or $p$.

The Challenge: While asymptotic estimates for such functions can be derived using sieve methods or Möbius inversion, obtaining fully explicit, closed-form exact formulas is difficult. Standard inclusion-exclusion methods yield piecewise periodic expressions involving alternating sums over divisors, which obscure the underlying affine structure of the counting function and are computationally inefficient for large $n$.

2. Methodology

The author develops a novel approach based on residue arithmetic and the Euclidean algorithm, avoiding traditional inclusion-exclusion sums. The methodology proceeds in four main stages:

A. Canonical Remainder Operator

The paper utilizes the canonical remainder operator $\delta_k(x) = x - k\lfloor x/k \rfloor$, which maps integers to their standard representatives in $\{0, \dots, k-1\}$. Key properties of this operator (compatibility with addition and multiplication) are established to handle residue constraints algebraically.

B. Minimal Solutions via Euclidean Algorithm

A core technical contribution is the derivation of an explicit formula for the minimal solution of linear remainder equations of the form $\delta_k(a_0 x) = b_0$ where $\gcd(a_0, k)=1$.

• Instead of using modular inverses directly, the author expresses the Extended Euclidean Algorithm entirely in terms of the $\delta_k$ operator.
• Theorem 3.2 provides a closed-form expression for the minimal solution $x$ as a finite sum involving sequences generated by the Euclidean algorithm. This allows for the direct calculation of specific residue classes without iterative search.

C. Decomposition by Residue Classes

The problem is decomposed based on the residue of $n$ modulo 3. Since $h$ and $k$ are coprime to 6, they must be of the form $6z+1$ or $6z+5$.

• If $n \equiv 0 \pmod 3$: One summand is $\equiv 1 \pmod 6$, the other $\equiv 5 \pmod 6$.
• If $n \equiv 1 \pmod 3$: Both summands are $\equiv 1 \pmod 6$.
• If $n \equiv 2 \pmod 3$: Both summands are $\equiv 5 \pmod 6$.

This reduces the problem to counting pairs of integers $(z_1, z_2)$ such that $z_1 + z_2 = m(n)$ (where $m(n)$ is a linear function of $n$), subject to the constraint that $z_i$ avoids specific residue classes modulo $p$.

D. Exclusion of Residue Classes

Using the minimal solutions derived in step B, the author identifies the specific residue classes modulo $p$ that must be excluded to satisfy the condition $\gcd(h, 6p)=1$.

• Let $a(p)$ be the minimal solution to $6x \equiv -1 \pmod p$.
• Let $b(p)$ be the minimal solution to $6x \equiv -5 \pmod p$.
  The counting problem is then reduced to counting pairs $(s, t)$ in a bounded interval that sum to a target value while avoiding the specific residues $a(p)$ and $b(p)$.

3. Key Contributions

1. Explicit Closed-Form Formulas (Theorem 4.13):
   The paper provides exact formulas for $g(2n, p)$ expressed using:

   • The canonical remainder operator $\delta_k$.
   • Elementary step functions (Heaviside-like indicators).
   • The minimal solutions $a(p)$ and $b(p)$.
     The formulas are piecewise affine, meaning $g(2n, p)$ behaves like a linear function along arithmetic progressions of $n$ determined by residues modulo $3p$.

2. Algorithmic Efficiency:

   • Precomputation: The parameters $a(p)$ and $b(p)$ can be computed in $O(\log p)$ time using the Euclidean algorithm.
   • Evaluation: Once precomputed, evaluating $g(2n, p)$ for any $n$ requires only $O(1)$ arithmetic operations.
   • Comparison: This is a significant improvement over direct enumeration (brute-force), which requires $O(n)$ operations.

3. Structural Insight:
   The work reveals that the counting function is not merely periodic but possesses a piecewise affine structure governed by the interplay of residues modulo 3 and $p$. This structure is transparent in the derived formulas but hidden in standard sieve-based estimates.

4. Results and Validation

• Theoretical Result: Theorem 4.13 unifies the counting process into two cases based on $\delta_3(n)$, providing a complete solution for all even integers $2n$ and primes $p \ge 5$.
• Computational Validation:
  • The formulas were implemented and tested against a brute-force oracle.
  • Scope: Validated for all even integers $2n \le 10^5$ and primes $p \in \{5, 7, 11, 13, 17, 19, 23\}$.
  • Outcome: Perfect agreement (350,000 test cases) was observed, confirming the correctness of the derived formulas.
• Visualization: The paper includes graphs showing the piecewise affine nature of the function, where parallel linear families correspond to specific residue classes of $n$.

5. Significance

• Computational Number Theory: The paper offers a highly efficient algorithm for counting coprime representations, shifting the complexity from linear in $n$ to constant time per query (after logarithmic precomputation). This is crucial for applications requiring high-frequency queries or large $n$.
• Analytic Clarity: By replacing complex inclusion-exclusion sums with explicit affine formulas, the paper clarifies the arithmetic structure of the problem, making the dependence on $n$ and $p$ transparent.
• Generalizability: The author notes that the method extends naturally to exclusion sets involving multiple primes $\{2, 3, p_1, \dots, p_k\}$, suggesting a pathway for solving more complex additive problems with multiple coprimality constraints.

In summary, Salazar's work bridges the gap between asymptotic estimates and exact counting by leveraging the Euclidean algorithm to derive efficient, closed-form expressions for a specific but fundamental problem in additive number theory.