Elementary closed-forms for non-trivial divisors
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arXiv
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| Format: | Preprint |
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2025
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| author | Prunescu, Mihai Shunia, Joseph M. |
| author_facet | Prunescu, Mihai Shunia, Joseph M. |
| contents | 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 $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_26939 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elementary closed-forms for non-trivial divisors Prunescu, Mihai Shunia, Joseph M. Number Theory 11A51 (primary), 11A25 (secondary) 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 $n$. |
| title | Elementary closed-forms for non-trivial divisors |
| topic | Number Theory 11A51 (primary), 11A25 (secondary) |
| url | https://arxiv.org/abs/2510.26939 |