Let $C_n$ be a cyclic group of order $n$. We prove that if $(n,6)=1$, then every minimal zero-sum sequence of length four over $C_n$ has index one, thereby resolving the length-four index conjecture. After the gcd reduction, the nonunit case follows from the theorem of Shen-Xia-Li, and the remaining unit case is solved by a new multiplicative Fourier argument. The index-two residue identity yields a character-moment relation, and the odd characters with vanishing first moment form an exceptional spectrum of size at most $157\varphi(n)/1440<\varphi(n)/9$. A finite-group uncertainty principle then forces the four-term multiset to be invariant under negation, contradicting minimality. Apart from standard facts about primitive Dirichlet $L$-functions, the remaining argument is finite and requires neither asymptotic estimates nor computational verification.
We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every...
For positive integers $n$ and $M$, Sundaram defined $ F_{n,M}=\sum_{\substack{d\mid n\\d\le M}} Lie_{n/d}[p_d] $ and conjectured that every coefficient in its Schur expansion is a nonnegative integer. We prove the conjecture. The power-sum expansion isolates the contribution of the identity class, and the proof divides...
Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities $1,2,4,\ldots,2^{m-1}$, normalized by $2^m-1$. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density a...
Let $X^4_m=\{x_0^m+\dots+x_5^m=0\}\subset\mathbb{P}^5$ be the Fermat fourfold of degree $m$. We give a computer-assisted proof of the Hodge conjecture for $X^4_m$ for every odd $m\le 199$: three geometric closure criteria combined with an exhaustive machine census of the Hodge $(2,2)$-orbits, with a completeness proof...
Let $\mathrm{IMS}_n(m)$ count the $n\times n$ nonnegative integer matrices whose row sums, column sums, main-diagonal sum, and antidiagonal sum are all $m$. We determine the Ehrhart series $F_n(q)=\sum_{m\geq0}\mathrm{IMS}_n(m)q^m$ as reduced rational functions for $n=7$ and $n=8$. Their numerator--denominator degrees...
We prove a nine-distance theorem for Kronecker sequences on flat three-tori. That is, we show that among the first $N$ orbit points, at most nine distinct positive nearest-neighbour distances occur. This proves the conjecture of Haynes and Marklof. An example of Dettmann shows that nine is optimal. More generally, we p...
N. Shulga· 4 citations· ⚡3
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
