Exact Formulas for Coprime Representations of Even Integers Avoiding a Prime

TL;DR

This paper derives explicit, efficient formulas for counting coprime representations of even integers avoiding a prime, enabling quick computation compared to enumeration.

Contribution

It provides closed-form, piecewise affine formulas for the function g(2n,p), involving elementary functions and minimal solutions of specific congruences, improving computational efficiency.

Findings

Formulas validated for all 2n ≤ 10^5 and primes p in {5,7,11,13,17,19,23}

Each evaluation requires only O(1) operations after precomputation

The formulas show g(2n,p) is piecewise affine along arithmetic progressions

Abstract

Fix a prime $p \ge 5$ and define $g(2n,p)=\#\{(h,k)\in\mathbb{Z}_{>0}^2 : h+k=2n,\; h\le k,\; \gcd(h,6p)=\gcd(k,6p)=1\}$. We derive explicit closed-form expressions for $g(2n,p)$ in terms of the canonical remainder operator $\delta_k(x)=x-k\lfloor x/k\rfloor$, elementary step functions, and the minimal solutions of the congruences $6x \equiv -1 \pmod{p}$ and $6x \equiv -5 \pmod{p}$. A key ingredient is an explicit formula for the minimal solution of $\delta_k(a_0 x)=b_0$ obtained via the Euclidean algorithm, which determines the excluded residue classes directly. The resulting formulas show that $g(2n,p)$ is piecewise affine along arithmetic progressions of $n$, governed by residue classes modulo $3$ and $p$. For fixed $p$, after precomputing two residue parameters in $O(\log p)$ time, each evaluation of $g(2n,p)$ requires only $O(1)$ operations, compared to $O(n)$ for direct enumeration. The formulas are validated computationally for all $2n \le 10^5$ and primes $p \in \{5,7,11,13,17,19,23\}$, with perfect agreement with brute-force enumeration.