Andres M. Salazar | Searcharxiv

arXiv2026

Restricted Coprime Representations Modulo 6P: Exact CRT Formulas and Positivity Thresholds

Let $5\leq p_1<\cdots 0$. Maximizing this bound over squarefree moduli whose prime factors do not exceed a prime $p$ gives an explicit threshold $\mathcal N(p)$ satisfying $\log\mathcal N(p)\sim(\log3)p/\log p$. We also describe the complete prime cutoff at $\sqrt{2n}$ 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 $n\geq13$.

math.GM↗

arXiv2026

Exact Formulas for Coprime Representations of Even Integers Avoiding a Prime

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 $δ_k(x)=x-k\lfloor x/k\rfloor$, 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 $δ_k(a_0 x)=b_0$ 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.

math.GM↗