Elementary closed-forms for non-trivial divisors

TL;DR

This paper introduces elementary closed-form expressions for finding non-trivial divisors of composite numbers, using fixed arithmetic operations, providing a novel symbolic approach that differs from traditional algorithms.

Contribution

The paper develops new closed-form formulas for divisors, including a simplified form that avoids factorial-unwinding and uses quadratic residue invariants.

Findings

Closed-forms can express various divisors with fixed operations.

Evaluation requires exponential time, but the number of operations is constant.

Contrasts with traditional algorithms whose complexity scales with input size.

Abstract

We present several elementary closed-forms that express a non-trivial divisor for every composite integer $n > 1$. Each closed-form consists of a fixed number of elementary arithmetic operations drawn from the set: addition, subtraction, multiplication, integer division, and exponentiation. Two families of closed-forms are developed. First, direct application of the hypercube method yields closed-forms $T_1(n)$, $T_2(n)$, $T_3(n)$, and $T_4(n)$ expressing the smallest prime divisor, largest non-trivial divisor, largest prime divisor, and greatest prime $\leq n$, respectively. The factorial-unwinding technique underlying these hypercube constructions leads to extreme symbolic complexity, motivating our main result: An alternative closed-form $T(n)$ that avoids factorial-unwinding by synthesizing the quadratic residue invariants $\chi(n)$ (largest $r$ such that $r^2$ is a divisor) and $\omega(n)$ (number of distinct prime divisors) with integer root extraction. Although evaluating these closed-forms requires exponential time, the number of arithmetic operations performed remains constant and independent of the input size $n$. This sharply contrasts with traditional algorithmic methods, where the number of operations required to locate a non-trivial divisor necessarily scales with $n$.