Abstract
Let p ≥ 5 be prime, and let g(2n,p) denote the number of unordered representations by positive integers 2n = h + k, with h ≤ k and 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 ↦ g(2n,p) is a degree-one quasipolynomial with 6p as a quasiperiod, and we give its exact increment under n ↦ 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 δ_q(x) denotes the least nonnegative remainder of x upon division by q. For p = 7, the formulas reduce to g(14n,7) = 2n when δ₃(n) = 0, and g(14n,7) = n when δ₃(n) ≠ 0.
Author ORCID Identifier
0000-0002-0889-7735
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
Salazar, Andres Mauricio
(2026)
"Exact Formulas for Restricted Coprime Representations of Even Integers,"
Communications on Number Theory and Combinatorial Theory: Vol. 7, Article 2.
Available at:
https://research.library.kutztown.edu/contact/vol7/iss1/2