Exact Formulas for Restricted Coprime Representations of Even Integers

Abstract

Let p ≥ 5 be prime, and let g(2n, p) denote the number of unordered representations by positive integers 2n = h + k, h ≤ k, gcd(h, 6p) = gcd(k, 6p) = 1. Every positive integer coprime to 6 belongs to exactly one of the families 6z + 1 and 6z + 5. In each family, divisibility by p excludes one explicit residue class of z modulo p. We use these two excluded classes to derive an exact formula for g(2n, p) from finite residue counts and nonnegative lift counts. The formula is valid for every positive integer n, including all boundary cases. We also prove that, for fixed p, the function n 7→ g(2n, p) is a degree-one quasipolynomial with 6p as a quasiperiod, and we give its exact increment under n 7→ n + 6p. Finally, we evaluate the formula on the subsequence g(2np, p). The form of the resulting expression depends on δ6(p) and δ3(n), where δq(x) denotes the least nonnegative remainder of x upon division by q. For p = 7, the formulas reduce to g(14n, 7) =    2n, δ3(n) = 0, n, δ3(n) ̸= 0. Keywords: additive number theory, restricted representations, coprimality conditions, exact enumeration, quasipolynomials MSC 2020 : Primary 11P83; Secondary 05A15, 11A07, 11B75