A counterexample to Sun's sign conjecture for cotangent permanents
Year: 2026
Type: preprint
Abstract: For each odd prime p, Zhi-Wei Sun proved that the normalized cotangent permanent t′p is an integer and conjectured that χp(−1)t′p < 0, where χp denotes the Legendre symbol.For p = 29, we obtain t′29 = 84,643,985,981,440 > 0. Since χ29(−1) = 1, the cotangent inequality in Sun's Conjecture 4.7(ii) fails.We determine the value by specialization to finite fields and the Chinese remainder theorem, using an explicit Archimedean bound to ensure uniqueness. A separate calculation in Z[X]/(1 + X + ⋯ + X^28) verifies the corresponding cyclotomic identity coefficientwise.Preprint version v1.0.0, 2026-08-26. Author and correspondence: Yuhang Shi, [email protected] Codex was used for literature screening, development and checking of the exact verification programs, and language editing. Responsibility for the mathematical claims and the final manuscript rests with the author.
Author: Shi, Yuhang
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204227923
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