On the cyclotomic field $\mathbb{Q}(e^{2\pi {\bf i}/p})$ and Zhi-Wei Sun's conjecture on $\det M_p$

Hai-Liang Wu; Li-Yuan Wang

On the cyclotomic field mathbb{Q}(e^{2π {bf i}/p}) and Zhi-Wei Sun's conjecture on det M_p

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arxiv 2401.05853 v4 pith:KWKIPX4A submitted 2024-01-11 math.NT

On the cyclotomic field mathbb{Q}(e^{2π {bf i}/p}) and Zhi-Wei Sun's conjecture on det M_p

Li-Yuan Wang , Hai-Liang Wu This is my paper

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keywords conjecturefieldcyclotomicmathbbzhi-weiarithmeticcertainconfirm

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In 2019, Zhi-Wei Sun posed an interesting conjecture on certain determinants with Legendre symbol entries. In this paper, by using the arithmetic properties of $p$-th cyclotomic field and the finite field $\mathbb{F}_p$, we confirm this conjecture.

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A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei
math.NT 2026-07 unverdicted novelty 6.0

Proves D_a(0)=0 iff p≡3 mod 4 and χ(a n!)=1, gives Pfaffian-square factorizations of the determinants for p≡3 mod 4, and settles Sun's conjecture when a=n!.