Elementary closed-forms for non-trivial divisors

TL;DR

The paper tackles the problem of obtaining a non-trivial divisor for every composite $n>1$ using elementary closed-forms. It first presents hypercube-based closed-forms $T_1(n)$–$T_4(n)$ that arise from factorial unwinding (via Wilson’s theorem) and then introduces a compact main term $T(n)$ that avoids factorial unwinding by combining the quadratic-residue invariants $\chi(n)$ and $\omega(n)$ with $m$-th root extraction to yield a fixed-length arithmetic term. Although evaluating these closed-forms is exponential in input size, the number of arithmetic operations is constant, illustrating a theoretical boundary where nontrivial divisors can be extracted in closed form despite symbolic complexity. An alternative $U(n)$ is also given, and extensive appendices provide explicit constructions and verification code, highlighting the distinction between fixed-length symbolic formulas and practical factoring algorithms.

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 $χ(n)$ (largest $r$ such that $r^2$ is a divisor) and $ω(n)$ (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 $n$.

Theorems & Definitions (45)

• Remark 1.1
• Theorem 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1
• proof
• Corollary 3.1
• proof