Restricted Coprime Representations Modulo 6P: Exact CRT Formulas and Positivity Thresholds
Abstract
Let 5≤ p1<⋯<pr be primes, put P=p1⋯ pr, and let gP(2n) count the unordered decompositions 2n=h+k for which both summands are coprime to 6P. We derive an explicit finite inclusion--exclusion formula for gP(2n) by parametrizing the admissible residue classes and combining the local restrictions through the Chinese remainder theorem. The associated complete-modulus correlation is a specialization of Nagell's totient function and determines both the finite local density and the affine increment gP(2(n+6P))−gP(2n)=κP(n). A parameter-density decomposition gives a residue-sensitive positivity criterion and the uniform sufficient condition n≥ 3(3r−1)∏i=1rpi/(pi−2)+2Longrightarrow gP(2n)>0. Maximizing this bound over squarefree moduli whose prime factors do not exceed a prime p gives an explicit threshold 𝒩(p) satisfying log𝒩(p)∼(log3)p/log p. We also describe the complete prime cutoff at √(2n) and its exact additive-prime interpretation. We also evaluate gP(2tP) explicitly using the real character of period $3$. The positivity thresholds are elementary and fully explicit; lower-bound sieves provide smaller asymptotic scales, and the present thresholds do not establish positivity at the complete cutoff for all n≥13.
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BibTeX
@misc{salazar2026restricted,
title = {{Restricted Coprime Representations Modulo 6P: Exact CRT Formulas and Positivity Thresholds}},
author = {Andres M. Salazar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.22305},
eprint = {2609.22305},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22305},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Restricted Coprime Representations Modulo 6P: Exact CRT Formulas and Positivity Thresholds
AU - Andres M. Salazar
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2609.22305
UR - https://arxiv.org/abs/2609.22305
AB - Let 5≤ p_1<⋯<p_r be primes, put P=p_1⋯ p_r, and let g_P(2n) count the unordered decompositions 2n=h+k for which both summands are coprime to 6P. We derive an explicit finite inclusion--exclusion formula for g_P(2n) by parametrizing the admissible residue classes and combining the local restrictions through the Chinese remainder theorem. The associated complete-modulus correlation is a specialization of Nagell's totient function and determines both the finite local density and the affine increment g_P(2(n+6P))−g_P(2n)=κ_P(n). A parameter-density decomposition gives a residue-sensitive positivity criterion and the uniform sufficient condition n≥ 3(3^r−1)∏_i=1^rp_i/(p_i−2)+2Longrightarrow g_P(2n)>0. Maximizing this bound over squarefree moduli whose prime factors do not exceed a prime p gives an explicit threshold 𝒩(p) satisfying log𝒩(p)∼(log3)p/log p. We also describe the complete prime cutoff at √(2n) and its exact additive-prime interpretation. We also evaluate g_P(2tP) explicitly using the real character of period $3$. The positivity thresholds are elementary and fully explicit; lower-bound sieves provide smaller asymptotic scales, and the present thresholds do not establish positivity at the complete cutoff for all n≥13.
ER -